Multi-slide animations in Keynote
Here's a self-answer, which I didn't know about when I wrote the question. Keynote has a "magic move" feature, which allows objects to move from initial positions on one slide to final positions on the next. This is a bit more limited than using action builds, because you don't have control over build order. (Objects always all move simultaenously, and the moves must all take the same duration.) However, you can use it in combination with action builds, in which case the "magic move" transition will animate the objects moving from their final positions on the first slide to their initial positions on the second. 

You can also simply chain 'magic move' transitions together, building an animation up out of multiple slides. In some ways this is easier than building the animation up out of actions, though it does have the disadvantage that if you want to make a change you have to manually change every slide in the animation.

I've found that with a bit of careful thought I can achieve all the effects I want in a fairly convenient way.

To use this feature, you have to use CMD-D to duplicate a slide, then move the objects to where you want them on each slide, and then select "magic move" as a transition effect between the two slides. For more details, see the link above.