What tools do designers use to produce usage animations when a mobile app is still in infancy (i.e., mockups, composites, prototyping)?

I would suggest a clickable prototype rather than video. You can then easily screen record a walkthrough of the app to a video but it's so much more compelling and convincing to put your phone in someone's hands and let them click.

FluidUi is a great rapid prototyping tool that is very easy-to-use and is pretty much drag & drop to make interactive (clickable) prototypes.

I've used it to demo to customers for feedback and raise funding.

There is also HotGloo and Proto.io (neither of which I've used).

I'm a big fan of InVision for clickable prototypes. You can string together static screens with hotspots that can look and feel like a mildly interactive site depending on the level of fidelity you're looking for. I've built many demos using this service. Clients respond well to the interface and developers understand what happens when you click different links. Let me know if you'd like to know more.
http://www.invisionapp.com/

As for quickly mocking up a video, you can hack together a quick version using Powerpoint (set timed transitions for slides) and add music or voice overs using iMovie.

Quicktime also has an out of the box screen recording option that can supplement the above options.

Above all, be sure you have use case defined in a script with linear flows articulated via screens.

Scott

Why not go one step further and produce something that can actually be used? My team uses our own frameworks to prototype in HTML/CSS, but when you don't have that I'd say the next best thing is Proto.io (http://proto.io). You can create interactive prototypes to test directly on the device in minutes without code.

If you really need video of interactions, I'm sure you can capture that from what you create with the tool. Even better, now you also have something tangible that you can hand to a potential investor or user to let them experience for themselves.

I'm not affiliated with Proto.io in any way. Since I prototype with code, I haven't actually used the product yet, but I've heard many great things about it. Hope this helps.

Basically if there's nothing super dynamic in there like a search field that needs to look like it's real - just use invision.

Axure is great when you want to take it out for user testing etc but overkill for a product video.