It is not a problem to fire .375 H&H in an improved chamber or in the .375 Weatherby as this is how you make your brass and is a process called "fire forming" The same applies to most improved chamberings such as the "K" Hornet. Your brass is made by firing factory ammo in the improved chamber. This is not the situation when you are firing a shorter cartridge in a longer chamber. At best you will eventually develop a ringed chamber causing hard extraction problems and basically ruining the barrel. Depending on how the cartridge is head spaced the worst case would be a blown up rifle an shooter.
The short case causing a ring in the chamber will even occur in guns such as .357 revolvers when enough 38 special ammo is fired over the years even though this is common practice it is not the best for the gun.