Welcome to the Grumble, framesalot.
You didn't supply much specific information about your project.
1. How deep is the rabbet in this frame?
You may be able to install some stiffer material than foamcore if there is enough room.
If you have enough depth in the frame rabbet and don't want to rebuild the frames, I wonder if Gater board wouldn't help to keep things in alignment. If the frames are warped WAY out of flat, you may have no choice but to rebuild them.
2. I am assuming that the frame and entire package have warped. Or has only the print and mats buckled?
If the frame is buckled badly, be aware of the fact that the glass may not fit into the rabbet correctly. And that won't bend very far if the frame is twisted badly.
3. Is there any possibility of rebuilding the frames out of new moulding and sealing the backs of the moulding with water based urethane?
It may not cure the problem but it could help to minimize warp in the future.
4. Are these frames old or recent? Old and not available anymore, you have no choice but to use what is there or change to a different profile.
5. Are they in a home near the beach or a large lake?
Wally Faye recently posted a good thread on what happens to framed art when it is hung in a beach front condo or home. Constant sea breezes, wet with mist, is worse than hanging art in a bathroom! The exposure to moisture is 24/7 in a beach front home and even the best framing may not hold up long under those extremes.
Just some questions to clarify the problem you face.
Framerguy