It is some small number of years into the future. We are regularly using beamed energy, as defined in this answer: large arrays of microwave transmitters. We have developed this so that these arrays are now positioned in orbit around Earth (powered by fuelled engines, regularly refuelled by probes).
Our space programs, however, have not developed that much. We're still stuck using liquid and solid fuelled rocket motors to get things into space. The one thing that has changed is that we now have access to an ion drive using electrostatic ion thrusters.
Our civilization has realised that in the not-too-distant future, the planet is going to die. They want to get out of there, as far away as possible - in contrast to the linked question, the Earth is going to explode so getting a long distance away is a good idea.
The problem with our ion drive is its energy use, which is high. We could just take huge batteries but we'd rather not.
So the problem I pose is this: assuming a sufficiently-sized competent team left on Earth to operate the beaming satellites, how much will the satellites help us? At what range are they no longer effective - and most importantly, can we get far enough away to avoid an Earth-destroying explosion?
This is hard-science. I'd like to see figures and equations where possible, but note if something cannot be calculated.