It seems like a decent attempt to deal with this sort of crisis is to think about what features in your new ontology cause you to label something as X in the old ontology.

For example, to recover values in your macroscopic physical continuity example...

For having mostly-the-same values:

Find the cognitive algorithms that cause you to label something as a macroscopic physical entity. Describe them in your new ontology in a way that lets you figure out if, within your new ontology, you would have labelled something as a distinct entity in your old ontology.

Probably more dangerous alternative:

You could try to figure out what about physical continuity it was that you valued, and what caused you to think that you valued it. If upon that reflection you decide to keep it, keep it.

This isn't a complete description of how to deal with it, nor is it like, mathematically formalized, but I'm curious as to what people think.