Yes, they do get moved around by their bishops. Long past are the days when a priest was named pastor of a parish and remained there until he either retired or died (the one who baptized me was in our parish for 38 years).
I think the bishop looks at what each parish needs, based on finances, pastoral care, etc and tries to place the best priest for the job in a community where he’ll do the most good.
Nowadays, it’s not easy, as we watch the decline in the number of priests, the demographic changes in parishes and priests who, in spite of having promised to obey their bishops, are very vocal about not wanting to be in a particular parish. Nothing can create more damage in a parish than a pastor who doesn’t want to be there and makes it very obvious. I lived through 4 years of that and it was disastrous.