My wife and I are testing a Don Hume IWB clip-on holster. It has been dyed black. The black dye does not come off on our clothes, but we don't sweat a whole lot up here in the Northwest.
Generally speaking, a properly dyed and sealed piece of leather should not "bleed" onto your clothing. If the leather has been oversaturated with dye, or of the leather's surface hasn't been sealed, waxed, or burnished, it will "bleed."
The best thing you could do is to write the the Don Hume people and ask to be reassured. Ask for written assurance (a guarantee) that their dyed leather won't "bleed" onto your clothes, or, at least, a statement from them about what conditions they expect would cause "bleeding."
People who do high-quality leatherwork should not be troubled by your question.