I've found that healers don't relocate at all just to heal people. If the enemy they are targeting is within their range then they won't move. If someone within their range needs healing then they will heal them (usually). But if you step out of their range then they will completely ignore you and just shoot the mob.
This. My healer will stand there and shoot, no matter how far away I am (I have all of his attacks disabled and in healing mode), unless I manually order him to heal, then he will start healing me again.
Note: I never have a pet out.