Use the Liquify effect and push the parts you want to move in the direction you want to move them. Doing this on a layer set to 50% opacity would help previsualizing the effect.
Then animated the Liquify Distortion percentage from 0 to 100%. Duplicate the layer, split it in half and put the first half at the end and the second half at the beginning of the comp.
Animate the first half to fade out and the second half to fade in.
Basically you’re repeating that liquify animation over and over, fading it in between.
—
For the next level of difficulty, animate the effect in both directions to blend it even better.