Hopefully this is helpful for at least some participating in this thread.
Costs associated with acquisition of the property such as title transfer fees for example, are capitalized, meaning they get added to the cost basis of the property and depreciated over time.
Cost associated with acquisition of the loan are amortized and deducted (not depreciated) over the life of the loan. So when you have a refi any amortized costs not fully deducted yet on the first loan are fully deducted in the year of the refi. Then your loan acquisition costs for the refi loan are amortized and deducted over the life of that loan.
So when you have a refi, there are no property acquisition cost associated with the refi loan. Therefore the cost basis of the property does not change and depreciation of the property continues on "as if" nothing happened. (because nothing did happen with the land or structure)
EDIT: 4/7/2019 to provide additional clarity
If your refi is with the same lender as the prior mortgage, then your amortized cost on that prior mortgage that have not yet been deducted, are added to your amortized cost of the new mortgage and the total is amortized/deducted over the life of the new loan.
If your refi is with a different lender, then any remaining costs left to be deducted on the old mortgage, are fully deducted in the year of the refi with the new lender. Then the "new" loan acquisition costs with that new lender are amortized and deducted over the life of the new refi mortgage.
