It depends (there's a lot more than X and Y in the animal kingdom), but broadly and loosely yes. It tends to be long periods of relative stability punctuated by rapid gene loss/transfer, followed by stabilisation of the smaller chromosome. The suppression of recombination that allows for a strong and stable sex division to develop also kind of makes that chromosome less stable. Y and X were probably, originally, hermaphoditic chromosomes that then had one develop a dominant male allele and then to stabilise it developed the SRY gene to regulate it.