I've got 20 videos with the same static background. Unfortunately during filming the camera was nudged so the background doesn't line up exactly across all the videos. Is there a way I can automatically crop, rotate and move the videos so the videos lineup precisely?
20 videos is not very many videos, and if the nudge happened between takes rather than during takes, the best approach is to align them manually, either by using the difference mode of a compositor or setting the opacity of each video to 50% and nudging them until all the backgrounds line up.
If you had 20,000 videos, I would suggest starting with Hugin, a tool for processing HDR and panoramic photos. The HDR problem concerns aligning multiple images so that a tone-mapping can be applied. The alignment heuristics for HDR would also align your videos (especially if you only compute the alignment parameters for the portion of your videos that are reliably background). You could play around with parsing the output of the Hugin control point finder.
This isn't specifically Premiere-oriented, but I would concatenate all of the videos into one and use the MELT framework with video stabilization. You might need to play around with the parameters a bit, so I would suggest cutting a 1min section with the nudge in the middle in order to fine tune your settings before running it over the entire video.