Can I take my address range with me when I change ISPs (i.e. renumbering)?

If the customer has PI addresses the assignment will remain valid providing the criteria for the original assignment are met.

Customers with PA addresses are recommended to obtain a new address space assignment from another LIR and return the PA address space that was assigned by their original LIR. In this way full use of aggregation is made. When a customer renumbers into an address allocation held by another LIR, their assignments can be routed as part of that LIRs allocation. At the same time the aggretability of the original LIRs allocated block is maintained.

It is the responsibility of an LIR to make it clear to their customers which type of address space is assigned to them.