How do I animate a property that my framework and libraries don't support animating

Over the course of working in different companies I've come across many different ways that frameworks and libraries animate things. Sometimes I'll say, "Man, I'm so happy I can animate a color with this library; but how do I animate rotation, or the alpha channel, or the clipping mask, or a text string, or anything that this library doesn't support?" So my question is, what do all GUI animations have in common? And How could I animate anything in the GUI?