As part of the loan approval process a lender not only approves you, the borrower but also approves the property. Property approval is performed with an appraisal of the property. The appraisal is an independent evaluation of the current market value of the subject property. And you gave the lender $500 for the appraisal. You should be able to transfer still use that appraisal at your bank now, right? After all, you paid for it.
Even if you paid for an appraisal the appraisal doesn't belong to you. Yes, you get a copy but it's not yours. In fact, if you look on the front of your copy it will state that the appraisal was prepared for your old lender. And your new lender isn't obligated to use it.
A lender can have its own internal guidelines regarding appraisals and one of those guidelines might be that they don't accept appraisals performed for other lenders. In this case you may be on the hook paying for yet another appraisal.
Some lenders do accept an appraisal to be transferred under certain conditions. First, the old lender will prepare some paperwork authorizing the transfer and the new lender will want to review the transfer paperwork as well as the appraisal itself. If the lender accepts the new appraisal you may only encounter a negligible fee called a "retype fee" which is nothing more than replacing the name of the old lender with the name of the new lender on the appraisal. If you change lenders midstream, make sure you're clear on transferring an appraisal. It's completely up to the lender to accept or deny a transfer request.