Can doctors ask for your mental health records to be sent to them?

Only if authorized. Only if you sign an authorization called release of confidential information that allows the holder of your records to whoever you specified on the form. Be aware that insurance companies will have you sign releases to give you coverage. Some exceptions to this are: releases as required or permitted by law such as court orders, and to avoid serious threats to self or others' health or safety.
With Permission. You must first give written consent for such information to be transferred from one physician to another, unless this transfer is done under court order. Your current physician cannot transfer this information unless it is within the context of direct care. The purpose of the request is also important, and you can release only parts of the information as requested.