There are many reasons for why a domain registrant might transfer a domain name away from its current registrar. By the time the transfer has been initiated, it’s most likely too late for a registrar to *save* the customer / domain name. There should be no reason for a registrar to delay a transfer.
Some registrars take up to a week to release a domain name. They offer no mechanism for the customer to approve the domain name transfer, and it moves forward at whatever pace is set by the registrar.