Does Bootstrap have any system for doing 'character/cutout' style animation?

It actually sounds similar from 2 years ago.

But understanding what your GIF’s look like would go a long way regarding suggestions if you’re still having trouble moving on after those 2 years. For anything elaborate or unique custom coding will be required as you can only get so much from any “preset” based approaches.

All of Chris Gannon’s work (seen in the link provided by @AllMediaLab above) is using custom coding and generally leverages paid Club GreenSock plugins such as DrawSVG, morphSVG, etc.