Mama Murphy won't switch settlements
You can't*. Character is essential for Sanctuary location. Especially dialogues are related to Sanctuary.
When you meet her in Sanctuary, she says:
"You're tied to this place, kid. Your energy"
That will be little off if you move Mama Murphy somewhere else like Graygarden where player never was before war.
*Maybe there is workaround with mods/console. But not standard way to do this.