44 Magnum is quite a bit smaller in diameter at the base than the 444 Marlin,,.012/.013" on spec,,could be a little either way with actual loaded rounds and different rifles.
The 444 is actually a rimmed Mauser x57mm case with a gentle straight taper to the 44mag neck diameter if you want something to compare it to.
The base diameter ahead of the rim is the standard .470" of the 30-06 & Mauser family of cartridges.
Rim thickness of the two is the same or within a couple .000".
Rim diameter is the same on both I believe,,that of the 44Mag case . Gone from the 444 was the large rim of previous lever action rounds.
So the 44Mag slides right in the 444 chamber and appears to headspace OK. The problem is the undersize case of the 44Mag in the 444 chamber.
It may just expand upon firing to take up the extra space,, or it may crack the side walls of the brass on firing. Depends on the individual 44Mag rounds in most instances.
The 444 is service pressure rated above the 44Magnum,,but don't take that as a license to use the wrong ammo in the chamber. You're dealing with 35K+ PSI any way you look at it.
Are they intended to be interchangeable?,,no.
Will a 44Mag fit in a 444Marlin chamber and go bang when you pull the trigger??,,yes.
I think the "...in a pinch..." in the OP's post is the key to this one.
Just be aware of what is going on.
There are alot of such combinations out there, some more dangerous than others.
JM .02