You can shoot whatever you like I guess, but the 44 Mag to my understanding was built around the 240 grain bullet. Many people agree it's the best all around bullet for the 44 mag. People have been shooting heavy for caliber bullets for some time now with good success on bigger game.
240,s seem to be fine for Deer size animals,
while many favor the 280/300 grain + for larger and dangerous game.
It doesn't mean the 240 grain can not do the job, the 300,s just do it a little better with more penetration.
If you like a lighter bullet for whatever reason and do well with it them, use them . I think you will find the for long range shooting the heavier bullets at sub sonic speeds maintain energy and velocity better than the lighter counter parts.
I use 265 grain bullets that i cast myself at about a 12/14 hardness.
I try to keep them about 1150/1200 FPS and they shoot through any game I may encounter.
For cans, paper targets and an occasional stick on the ground I don't think it matters any at all, why would it ?
Availability has entered into the what should I shoot now game.
You can't shoot what you don't have----unless you cast your own.