A firearm chambered in .44 magnum (.44 Rem Mag) will shoot .44 special and .44 magnum.
I don not believe that the .44 WCF cartridge will fit in a .44 chamber.
If it will then the firearm might techically be able to fire a 44WCF (.44-40) cartridge but it could be dangerous and inaccurate. The 44WCF has a bullet diameter of .427 and the .44 Mag has a bullet diameter of .429. The .44 special and the .44 Rem Mag are straight cases while the 44WCF cartridge is a bottle neck.
The .44-40 case has a diameter of approx. .469" above the rim, tapering to .457" at the shoulder with a .443" od. neck while the .44 Special/Mag is a straight-wall .457" from rim to case mouth. In short, they won't swap.