I've been repeatedly asked to turn this into a PSE question, so here goes.

Some people may have noticed that, when I answer a grid-deduction question, my answer often includes, as well as a step-by-step explanation of the deductions and a picture of the final solved grid, an animated GIF showing the deductions being entered into the grid one by one. (Here's a couple more examples in case my explanation isn't clear enough. Self-promotion alert, obviously.)

How are these gifs created? (Obviously I have an answer in mind, but I'm also open to other answers if anyone else has a better method than mine.)

I think this question is on-topic since it's about how to show the solution process of a puzzle, which is similar to what we often do in the puzzle-creation tag. (I don't want to create a puzzle-solution tag as it'd probably be dreadfully misused.)