When your Crape Myrtle gets too big for where it's planted... Can you transplant it? And if you can, When is the best time?

Yes, you can transplant it but, as with all/most deciduous plants, it's best to wait until they're dormant (lose their leaves in the winter). If yours is becoming too large for your area and depending on where you live -- did you know that you can prune crape myrtle into a tree form? That might solve the too little space problem with all the greenery up high. I once saw a 20' tall crape myrtle at the Orange Show grounds near San Bernardino, California. It was breath-taking!