If you took a deciduous tree from the northern hemisphere and transplanted it in the southern hemisphere,?
Yes, of course.
It would upset whatever cycle the tree was in. But if you took it from autumn and transplanted it in Spring, it would presumably burst immediately into growth. Taking it from Spring to autumn may kill it, because it would miss a complete growing season.
I think they would adjust fairly quickly to the new daylight and temperature schedule. I cannot say I have ever seen any write-up of such an experiment, but I would think that if such transplantation would fail, it would be a well-identified problem and in the general knowledge realm. I do know that simply moving a plant (trees, saplings) from the outside to the inside in the fall can retard the leaf die-off, because I have done it.
You need to consider what it is that controls the leaf cycle.