Although it's not specified in the game, I don't believe Dinopolis was pulled from a point millions of years in the past, but rather from a point around the same time period in a different dimension - or timeline.
I'm still not really sure about the precise sequence of events in Chrono Cross, even after multiple playthroughs. Let me list the things that I know of and try to make sense of it. (in the context of canon as much as possible)
1. In the year 65,000,000 B.C. Lavos crashes into the earth, creating an ice age that causes the extinction of the Reptites - the precursor to the Dragonian race. Despite this event, the Dragonian/Reptite timeline continues in another dimension - one in which Lavos never crashed into the planet. (This suggests that the Lavos event was a temporal anomaly, rather than a destined event, and that the Dragonian timeline was the original history of the world.)
2. In the Dragonian timeline, Reptite civilization evolves in a way that is highly attuned to nature. The Dragonians create Dinopolis at some point and create an artificial plasma life-form (the Dragon God) to help govern the natural energies of the planet.
3. In the Lavos timeline, human civilization is not destroyed by the Reptites and is able to develop freely. At some point humans learn the ways of magic, and the Kingdom of Zeal is founded.
4. Around the year 12,000 B.C., the Zealians discover the power of Lavos slumbering deep under the ocean. An undersea palace is constructed, and Queen Zeal's mind becomes influenced by Lavos. Hoping to harness the incredible energies of Lavos, the Mammon Machine is constructed. Princess Schala is instrumental in the building and/or operation of the device.
5. Time-travelers (Crono and company) come to Zeal in their quest to track Lavos down in their past and destroy it. Their first true encounter with Lavos ends in disaster when the Mammon Machine is activated by Queen Zeal. Schala manages to save the time-travelers, but in so doing becomes a prisoner of Lavos. Knowing that the time-travelers represent an existential threat to itself, Lavos possesses Schala and sends her to the Darkness Beyond Time as a contingency against death. With both Queen Zeal and Schala in its thrall, Lavos then destroys Zeal. Belthazar is sent through a temporal distortion to the year 2300 A.D.
6. Crono and his companions manage to destroy Lavos in their second encounter. With the destruction of its body, Lavos' essence transfers fully into the mind of Schala - in a place where time-travelers cannot easily find her. Schala is driven mad by the contact with the alien consciousness and begins to despair. Schala's brother Janus finds her in the Darkness Beyond Time but is unable to free her. Janus' memory is erased.
7. Instead of a ruined future, Belthazar suddenly finds himself in the highly advanced world of 2300 A.D. after Lavos' body is destroyed. He quickly assumes a position of leadership in the scientific community.
8. Discovering unusual magnetic activity in the barren El Nido archipelago, Belthazar builds a temporal research center called Chronopolis in the Sea of Eden. (The unusual magnetic activity is believed to be caused by the Dragon God, which is located in the same region in the Dragonian timeline.) The facility is powered by the last surviving fragment of Lavos' physical form, the Frozen Flame, which has the power to transcend space/time.
9. Belthazar discovers Schala's fate and perceives that the combined Schala/Lavos lifeform will consume all of space-time if Schala's sanity is not restored. He then conceives a bold and convoluted plan to rescue her: "Project Kid". The plan finalized, Belthazar leaves instructions for the Chronopolis staff and then disappears. (At some point, Belthazar and/or Lucca creates a failsafe to prevent the FATE supercomputer from turning against them in the form of Prometheus. It is not at all clear how much Lucca is read into Belthazar's plan, if at all.)
10. Following the instructions left by Belthazar, temporal experiments continue in Chronopolis for the next 100 years. In the year 2400, the culmination of Chronopolis' research is realized, and scientists attempt a "counter-time experiment". (It is not clear what this means, but it is suggested that its success will enable Chronopolis to control time at will - perhaps indicating an ability to alter the timeline without directly affecting the reality in which Chronopolis exists, thus negating the change.) The safety interlocks on the Frozen Flame are reduced to minimal levels and then all hell breaks loose. The objective of the experiment is not made clear, but what is (somewhat) clear is that the Frozen Flame senses the power of Lavos destroying Zeal in the year 12,000 B.C. and attempts to join with it. The entire Sea of Eden is then encased in a temporal bubble and sent 10,000 years back in time before the Chronopolis staff can shut the experiment down. The sudden temporal shift of such a large area creates a huge temporal distortion that effects not only the dimension of the humans, but the dimension of the Dragonians as well - and even the Darkness Beyond Time, where Schala exists.
11. Due to the massive temporal disturbance in the Sea of Eden, several things happen at once. The Dragonian city of Dinopolis, and the "Dragon God" controlled by it, is ripped out of its home dimension and into the dimension of the humans - and is thrown back in time to the same moment that Chronopolis emerges in. During this transit, the Dragon God briefly crosses into the Darkness Beyond Time and makes contact with the Lavos/Schala entity. This contact momentarily frees Schala's will from Lavos' influence and she suddenly becomes aware of the agonizing cries of a young child through the distortion. Schala then forcibly uses Lavos' transcendent powers to attempt entry into the year 1006 A.D. to save the boy and possibly escape herself. (This action creates a fierce storm in that time-period, which enables Serge to reach Chronopolis and be saved by the Frozen Flame.) Although Serge is saved, Schala is unable to break free from her prison, and she, Lavos, and the Dragon God combine into one wrathful entity called the Time Devourer.
12. The humans of Chronopolis and the Dragonians of Dinopolis at first regard each other with curiosity, but relations quickly break down when the Dragonians determine that the humans originate from a "false reality" and launch an attack on Chronopolis. Due to the power of the Frozen Flame, and in no small part to the unexpected corruption of the Dragon God by Lavos, Chronopolis prevails. Taking the powers of the Dragon God for their own, the humans of Chronopolis conceive a plan to preserve their timeline in spite of their unexpected, and irreversible, backward jaunt through time. The El Nido isles are created over an unknown number of years, and are then colonized by the descendents of the Chronopolis staff under the guidance of the FATE supercomputer. (It is not clear when the colonization takes place, but the year 900 A.D. is suggested.) Special terminals are created to interface with FATE so that the populace and any visitors can be controlled in a way that does not threaten the timeline of 2400 A.D. The Dragon God is then split into six distinct entities and sealed away on the islands. The resulting Dragon Gods retain the memory of their combined form, but are unable to resist the Frozen Flame's ability to confine them. With part of its essence in effect trapped, the Lavos/Schala entity becomes dormant in the Darkness Beyond Time.
13. In the year 1006 A.D. the electromagnetic storm caused by Schala in her attempt to rescue Serge momentarily disables FATE in Chronopolis. This allows Wazuki and Miguel to enter the facility carrying the poisoned Serge. When power is restored to the facility, the Prometheus program, analyzing the situation, briefly assumes control of FATE and guides Wazuki to the Frozen Flame - using the young Serge's retinal scan to establish a new "arbiter" protocol. After Serge is healed and exits the reactor chamber, FATE is effectively cut off from the Frozen Flame and can issue no new commands to it - greatly reducing its ability to respond to changing conditions in El Nido. (And thus threatening its very existence.) Becoming aware of the sudden change, FATE overrides the Prometheus program and attempts to capture Serge with robotic sentinels, but the powerful Miguel is able to hold them back, enabling Wazuki and Serge's escape - but not before Wazuki is infected by nanomachines.
14. Taking advantage of the power loss suffered by Chronopolis, the six Dragon Gods use their combined powers to create an offspring in the form of a human girl, hoping that she can avoid FATE's notice long enough to undermine the computer's mission. This activity by the dragons briefly wakes Schala in the Darkness Beyond Time, who, curious about Serge's fate, stretches out her senses across history.
15. Four years after his escape from Chronopolis, Wazuki is being driven mad by the nanomachines unknowingly injected into him by FATE. His will is steadily eroded until one day FATE is able to take full control of his actions. Angered at the chain of events that caused it to lose control of the Frozen Flame, FATE alters Wazuki's form into that of a feline demi-human; a form that would terrify the "arbiter", Serge. Conceiving a plan to override the arbiter protocol and restore full access to the Flame, FATE decides to simply eliminate Serge. Wazuki is thus transformed into Lynx, and the new living interface for FATE throws the seven-year-old Serge into the sea and leaves him to drown. Seeing this event unfold from the Darkness Beyond Time, Schala uses all of her will to create an offspring of her own, but for the express purpose of preventing Serge's death. She then sends the infant-clone to the year 1005, where it is immediately found by her old friend Lucca Ashtear. Lucca raises the child like a baby sister, but it's not clear just how much she knows about "Kid" or her role in Belthazar's plan.
16. Bringing his override plan to fruition, Lynx tracks down Lucca with the help of his new lieutenant, Harle. His objective is to force Lucca to use her intimate knowledge of Prometheus' core programming to override the lock on the Frozen Flame. It is not entirely clear what happens in the tragedy that unfolds, except that Lynx fails and Lucca doesn't survive. The ten-year-old Kid manages to escape the burning orphanage and swears revenge against Lynx.
17. In the year 1020, shortly after her arrival in El Nido, Kid is suddenly confronted by a strange portal in the water. The portal is a time-gate created by Schala. Compelled to jump through the gate by her genetic twin, Kid is sent ten years back in time where she rescues the seven-year-old Serge from drowning, then is compelled to jump back through the gate where her memory of the event is erased. This action effectively changes history, but because it was initiated in a world where Serge was already dead, that world would have no choice but to continue as it was in order not to create a paradox. A parallel reality is thus created where the rescued Serge is allowed to live and prosper, no one being aware of reality's newfound division. No one but the original FATE.
18. At the moment of Serge's rescue in 1010, a great calamity unfolds in the Sea of Eden in the new reality. Since the reality of Serge's survival did not exist before the year 1010, everything that happened before that time in the original world would have no bearing on the new world's future history. Called the "Armageddon-branch theory", the heroes that saved the original world from Lavos' future destruction could not possibly do so in the new reality since that series of events originated in a year where the new reality didn't yet exist. And since the destruction of the original world was destined without the intervention of the time-travelers who prevented it, the new reality's future was doomed to be destroyed by Lavos, even though this world existed on the same time-axis as the time-travelers who saved the original world. The result of this paradox is a "Time Crash" centered in the Sea of Eden. Chronopolis came from the year 2400 in the world saved from Lavos, but in the new reality the year 2400 was from a world destroyed by Lavos in 1999, so Chronopolis could not possibly exist. This paradox, combined with the giant temporal shift caused by the original world's Chronopolis, causes the space-time continuum to shatter around the Sea of Eden - freezing time in an unrecognizable moment combining aspects of 400 years of history between Lavos' destined day of destruction and the day of Chronopolis' "counter-time experiment". This area is thereafter called "The Dead Sea".
19. Unaware of the Time Crash in his world, Serge lives a remarkably ordinary life up until the day when Kid (in the original world) travels 10 years back in time to save his life. When this moment comes, Serge immediately falls through the dimensional divide while he is at Opassa Beach with Leena. He meets up with Kid later that day. (And from her point of view, she rescued Serge only hours earlier.)
Well, that's how I interpret canon at the beginning of Chrono Cross, with a little supposition.