Of course it's better to have them moving separately, otherwise somebody would move 12*n times the month to move n times the year, before discovering that the year also moves, if they'll ever discover it. However a better solution would be to show only the month and only the year in two separate screens. No surprises, no chances of using the UI inefficiently.

And for 12 months, you can show all of them together on the screen. Same for the years, you can show more than one at the same time and scroll many of them together.