Your KFs will definitely appreciate a sunnier spot. It will increase the rate of growth and produce a sturdier plant that will not flop as much.
Technically.... since KF is a cool season grass you can pretty much move them anytime the ground is not frozen... though I prefer not to move grasses or any other kind of plant in the dead of summer if it can be avoided. The hot dry summer dog days can be very stressful for them and I dont feel like being a 24hr babysitter/waterer! The roots are pretty good sized like most grasses though not really enough to call tap roots. But you do need to get a decent sized rootball dug up when you are moving it... hopefully twice the size of the width of the base of the grass clump. If you can't manage that then make sure you get at least a few inches more all the way around.

Good luck!

Sue
"The one thing all gardeners share in common is a belief in tomorrow"