It's generally a one-time thing.

The systems are not really designed to "unpost" miles from one programme, and even if you managed to get one programme to do this, the next programme to which you go, and try to do a retroclaim, may refuse on the basis that they can see that the miles were already credited elsewhere (maybe even if this is just on the basis that the boarding pass you provide as justification shows that you had that other FFP number on the boarding pass). It may never be visible that miles were "uncredited", so you could find yourself disposing of your Avios/Tier Points, and then never getting anyone else - or even BA, a second time, if you went back in desperation having failed to get Qatar to post the miles - to ever credit those flights to your account again.

It *can* be done, but it would not be straightforward, it may not ever actually work (so you may lose credit EVERYWHERE for the miles), and you may find no-one who is willing to help you, and you would spend months of your life on phone calls. You would need two willing, able agents on both programmes and even then, it's unlikely to work correctly, or work first time, and you will find yourself banging your head against various brick walls.

Or you may just be point-blank refused. Given the current brou-ha-ha about BA's changes, I imagine many people are ringing up and demanding the same thing - so I expect you would get short shrift from BA for ringing up and asking them to do this.