I had this problem running firefox. It turns out that I had the snap version installed (according to which firefox). I installed the apt version with
sudo apt install firefox
and now firefox launches Firefox over MobaXterm just fine.
After discussion in the comments it turns out, that the problem can be reduced to detaching the X-Server from its shell window. There are a few ways to achieve this:
"onboard": Create a .vbs file containing
sh.Run "some.bat", 0
And run it with wscript
"onboard, cheating": Use the Task scheduler with ...