In the “Firefly” universe, astronomers discovered a possibly inhabitable multi-star system in 2020, with discoveries in the following years of dozens of habitable worlds.
In 2030, ways to alter and control gravity were discovered. That year, it was also announced officially that Earth’s biosphere would not support humanity within 100 years.
By 2048, technology had developed to the point that a mass evacuation to the new start system was deemed the only possible way for mankind to survive. They built enormous ships with tech that allowed them to reach one-third of the speed of light. It still took them over 100 years to reach the new system....but they did and humanity thrived.
In reference to your post, they didn’t develop warp drive. It was based more on the “Project Orion” tech than anything else.
Oh...according to the sources for the “Firefly” universe, the last exodus ship left Earth-That-Was in 2100. IN 2110, the last telemetry from Earth ceased...they assume that the population went to zero.
Or civilization succumbed to Green Politics and everyone was living in the 19th Century.....................