No it really really really shouldnt, thats what you got custom characters for. The Origin characters are the classes tyhey are, because well its intrinsic to who they are as people, their background and lifestory. Its also baked into their dialogue and who they are as a character. If you change Wylls or Shadowhearts class, they stop being that character, and 70% of their dialogue is wasted....

"They say he who smelt it dealt it."
Sooo technically... this burnt corpse is your fault officer."