If you are going m4/3 take some time to understand the system,there is a choice between Olympus and Panasonic,both systems are interchangeable as far as lens/body is concerned.
All mirrorless cameras use a contrast system for AF (apart from some that are out of your price point) DSLRs use phase detect,contrast can match or beat phase for still subjects but phase will win for BIF.
To stay anywhere near your budget i can only talk about contrast models.
For still subjects AF you will find no difference between Oly or Pana,i know this is correct as i have both,Pana have added something called DFD to their focusing you may want to research it as it can help you use CAF,to get the same focus results with Oly you would need to use SAF and pump the shutter.
Olympus lenses offer no built in image stabilizing but bodies like the the EM10MK11 have body stabilization which is good for 4 or 5 stops though this does get less effective as you get on the longer end of zooms.
Pananasonic use lens stabilization with some bodies also haveing built in stabilization this can in some combinations used together to give dual stabilization.
Going to stop here and wait to see if what questions you have.
With Olympus i would suggest you look at the EM10MK11,but make sure its the MK11,the next model up is the EM5MK11 and it offers no benifits and perhaps its not as good in some ways.
Panasonic which i favour for useability is a bit more complecated as a few of the models in your price range could suffer from shutter shock with certain settings,this though can be avoided completely by useing electronic shutter,the down side would be it can cause distortion with vertical objects if your panning for BIF.
Plese excuse the mixed layout i used preview post a coulpe of times and its mixed the running order up.
Panasonic G80, 100-400