Not that I have noticed here. Fox squirrels used the be the primary one in this area and I have seen them jump from tree to distant tree branches with no hesitation. Probably the fox squirrels have figured out that the grays will knock enough feed down that they do not have to work.

Our conservation department has printed several articles that the two groups do not interbreed. However we rarely see a pure squirrel of either type. You might start seeing some every interesting combinations. The one I really liked was the longer fox fur on a white bellied fat one.

I see the fox squirrels jump off my feeders when my dog gets let out the back door.

Perhaps their size makes them less agile and less able to make the HUGE jumps like the grey ones, but they still jump pretty far in my yard to get on and off the bird feeder.