No where do I see mentioned how much was actually saved in dollar figures. What are we talking here—thousands, hundreds? Similarly, I'd be interested in knowing engineering effort involved in the migration.

As a software consultant focusing on designing, building, and deploying software on AWS, more often than not, the infrastructure cost issues I've seen have less to do with the underlying infrastructure provider — AWS in this case — and more to do with the application itself that's being deployed. Most recently, we were able to shave off 50% of a client's bill that was using Lambda for compute. The issue? Several, including sleep timers (on a service where you pay by the millisecond!) as well as pathological code (i.e. consume message from queue, enqueue same message, rinse and repeat).

Yes. You can save money by transitioning to another provider. But I'd start with reviewing the underlying code and architecture first