I contacted MobaXterm and it looks like this is indeed behavior that they see and it sounds like it isn't meant to be this way. Maybe (hopefully) this will get changed in a later update.
This was their response:
"Hello Daniel, I am from the support team. Thank you for reporting this behavior. I managed to reproduce it on my side. I will talk with the development team in order to see if we can improve this at a later release of MobaXterm. Best regards.
Question: The alt-f and alt-b seem to only work with the left alt, not with the right alt. The right alt combinations work with Xterm so I don't think it's a keyboard hardware issue.
Is there something I'm missing? Is there somewhere I can configure this to work?"