The Ballista can only fire while stationary, but what if it's stationary aboard a moving ship? For example, on a Starlifter, Kraken, or Idris?

The Ballista will still fire aboard a moving Starlifter or Idris, though we're not sure launching missiles inside a moving ship is the safest thing to do! As for the Kraken, there's no reason why you couldn't fire missiles from a landing pad. However, the chance of hitting a friendly ship or part of the carrier could be quite high, particularly if it's moving.