The effects of multiple shots (in rapid enough succession to heat up the barrel) varies with every barrrel, with one of the major factors being the barrel length, and thickness.
Target shooters prefer a "bull" or "varmint" contour- the "heavy barrel" moniker if you will. The greater mass of steel takes longer to heat up- and harmonics are also affected less by the heat, maintaining same POI even with a hot barrel. Shorter barrels also generate less "whip" and have an advantage from that perspective. Generally speaking, I would go with a barrel as short as necessary to deliver the desired muzzle velocity (which usually doesn't make as much difference as many think) and as heavy a contour as comfortable for the intended use.
Hunters OTOH need to be able to hit POA on the first shot- and prefer lighter barrels for obvious reasons.
As mentioned, clean/dirty and hot/cold have varying effects on barrels, and every one is different. Some guys clean both powder and copper fouling after every shoot, some only when accuracy degrades, and most- somewhere in between...