And here's how the shader nodes were configured. Each texture was loaded as a separate 2D image map, and plugged into the inputs of the blender math-node. The blender node itself is plugged into the diffuse channel of the material.

Note the Blending value. This determines the percentage of dissolve between the two textures. At 0.5, the result would be halfway between texture 1 and texture 2.

Since this is an animation, I needed the Blending value to gradually change over time, from 0 to 1. I plugged in some math and variable nodes as an experiment, but for something this simple it isn't necessary. You can keyframe the changes in value, just as you can keyframe movement. Just set the value to 0 on the first frame of the animation, and to 1 on the last.