I can’t think of an automatic way to do this, which isn’t to say it can’t be done. 🙂 Unfortunately the time offset feature in the effectors doesn’t influence material animation.
Here’s what I would do:
* create a material with multi-shader in the luminance channel;
* add a bunch of layers to the multi-shader and load the video into each one;
* go into the animation properties of each layer and manually change the start time of the clip;
* assign a random effector to the cloner set to randomize color only.
That will randomly assign the multishader layers you set up to the clones, so you will have as many different start times as you have layers in your multishader.