Verify the old loan is closed, the new principal is correct, all fees match the offer, auto-debit works and the first statement reflects the agreed benchmark and spread. A transfer is not finished when the new lender pays the old lender; it is finished when both records are correct.
Use the same-end-date, after-cost comparison as the final decision rule. A transfer should reduce total future cost with a comfortable margin, not merely produce a prettier EMI.