Would be possible to power an E-Ink screen with a mechanical wristwatch?

As I understand it, the "power" budget here, is to effect change. An eInk display is statically showing the arrangement of the beads.

So a mechanical watch would presumably be doing something to charge a capacitor, to get sufficient "power" in electrons, to push some beads around.

If you want to count seconds with more than 1 bead, I suspect no. If you want to count minutes with a single digit, as long as its a low bead count, maybe. if you were up at 5 minute granularity, I think its possible high quality changes could be pushed out.

You might be re-winding the clockwork a lot.

I don't know about a watch but a power supply using kinetic energy would work such as a small lever that you could move. Kinetic power supplies sound nice but they are usually way more expensive that a battery even after you factor in the life time battery costs associated with powering the gadget. Plus it's inconvenient. It's way more practical to have a general purpose kinetic energy power source.

Here's an article that will give you some insight.

https://en.wikipedia.org/wiki/Energy_harvesting#Human_power