^ The way I see it is something like this...
Several hundred years from now, if humans have evolved past the primal instincts and there is world peace, the populations will settle down to sustainable levels. Human civilization will have achieved an ecological balance. Technological advances will progress at a slower rate, but more efficient in terms of expended overhead and minimal impact to the environment. People will be living very long and comfortable lives.
But over the course of several hundreds years that follow, human beings won't be satisfied to just live "as is", unable to further expand within the confines of Earth without disturbing the balance of the ecosystem. Some people will endeavor to explore the oceans and create submariner cities. Others will be compelled to take a chance on long distance interstellar travel, once technologies make it possible. If cryogenic freezing technology ends up perfected, and space craft drive and power systems sufficiently efficient for being able to reach identified habitable worlds many light years away, "project propagation" will recruit volunteers for making the one-way voyage to the most suitable known planet. And who knows... along the way, we may eventually develop superior spaceflight capability and send a few more expeditions in the same manner. The whole idea is being able to seed humanity on other worlds in the galaxy and not wait for an apocalyptic "abandon Earth" scenario at the very end. This is how I think we'll ultimately end up spreading out to other worlds.
But you're again assuming that it's a choice between "stay on Earth forever and eventually perish" and "expand to other stars and other solar systems." Apart from the many options available on Earth alone, the rest of the solar system is within immediate range of colonization. "Project Propagation" is more likely to involve moving much of humanity's population off world to cities on the moon, Mars and the moons of Jupiter before they convince anyone to expand to other solar systems.
Nor is it likely to require or be preceded by world peace. It's far more likely that the first exodus from Earth will be carried out by a Chinese program as an attempt to relieve the pressure of its own exploding population, and the inevitable fears that China may use its offworld population as an excuse to annex whole solar system bodies would be the cause of political tension and/or war. By the same token, the first expedition to colonize other solar systems would probably be launched from some major off-world industrial power (say, the Confederated States of Greater Saturn).