The short answer: 'No'.
I work with animals quite a bit with (at one time)* Hassy H and X and GFX. The later versions of all MF cameras focus faster than their predecessors, but for capturing moving animals, no. Even rapid head movements of a stationary animal (to keep eyes in focus) would be a challenge. You'd be forcing the system to do something it was not designed for and the technology isn't ready for - yet. You could do it, but the hit rate will be very low unless under very controlled conditions and ideally w/a trained animal.
That will hopefully change as new cameras progress. I'd echo Rand47s' comments on acquiring a nice used FF body and a couple of lenses for that type of work.
*We LOVED MF files and sensor size for portraits, etc., but are moving to the Sony A1 system for the AF and tracking.