Hi Julia,
Here’s one way to go about it:
*set your cloner to grid mode and set up the appropriate number of clones;
*in the cloner’s “Object” tab, set “Fix Texture” to “Straight.”
*apply your texture to the cloner object and set up flat mapping to get the UVs in order.
That’s it. Now you can move the clones about with effectors and the texture will stick to the clones as they move.