Again, use Gimp.
Make as many images as the number of frames you want in the final animated picture. More frames will make the image big. Open the Layers dialog in Gimp. Put each image into a different layer. You can rename the layer by appending the time each frame has to be on before the next frame appears. Something like layer 2 (100ms). This means that the layer 2 will appear for 100 milli seconds or in other words a tenth of a second. After renaming all the layers this way, save it as Gif. There is an option to save as animated Gif. Also check the "optimize" option to get a small animated avatar. If the changes from frame to frame are minimal, the size after optimization will be small. On the other hand, if they are unrelated images, it will be quite big.