I've built a model to batch clip rasters. It uses iteration and works fine on machines running arcGIS 10. However, colleagues that have upgraded to 10.1 cannot open it. the red x appears with a "unable to execute selected tool" pop up. Below is a picture of the model (I can't export the script for some reason):
I would not suggest trying to run a model created in 10.0 in 10.1. When we were making the switch to 10.1, I made a copy of all my toolboxes, renamed them as "Toolbox_10.1", and then tested them. None of them worked immediately; most of them I just had to drag each tool in the model back in from ArcToolbox, make the settings the same and then delete the original process. A bit tedious, but it guaranteed that the models would work again.
As mentioned above, copying and pasting the tools from the old model to a new one does also work, but I prefer dragging in a "fresh" copy of the tool. I have corrupted several of my version 10 toolboxes simply by opening them in 10.1, so creating a backup is a good idea.