Which pitcher is responsible for the players left on base when the original pitcher leaves the game? For example, Pitcher A, is pulled out of the game with 2 men left on base. A reliever - Pitcher B - comes in and a run is scored. Is Pitcher A responsible for the men on base or the reliever, or pitcher B?
The pitcher who is responsible for putting the men on base is the one that the runs are charged against. In this case, the two men on base at the time the reliever came on are charged to the original pitcher, A. Runs for men put on base by pitcher B are charged to him.
It pitcher B came in while his team is leading by say, one run, and the reliever gives up a hit that scores one or two runs, he is charged with a blown save. If giving up the second run results in a loss (with no further runs scored in the game), the loss is charged to pitcher A,
The runners on base at the time of a pitching change are called inherited runners, and if any of them score, the run(s) are considered given up by the pitcher who allowed them on base (in your example that is Pitcher A).