Since you're not bashing Catholics...
Were he a Catholic priest caught like this would the Church simply transfer the pervert at this point? Not bashing the Catholics, just not sure what their position is at this point?
Nowadays, he would be immediately removed from any form of ministry. If he was reported to the cops, the cops would deal with it. If he was reported to the Church superiors, they would investigate and if there was any possibility of it being a legitimate complaint, the Church would turn him over. The way the rules are written now, he is basically guilty until proven innocent. See here for the norms for handling these cases (even though they say that the accused has the presumption of innocence, you can be sure that they are defensive enough that if there is a question, they will err on the side of caution).
But I think you are asking about the situation back in the late 60s and 70s (when most of the abuse happened).
If the pervert was reported to the cops by the parents, then the legal system would take charge. Mostly, though, that didn't happen.
In the majority of cases, the pervert would tell the kid that he was bound to secrecy and would go to hell if he revealed any of what happened. That is why the majority of cases weren't revealed until after the revelations in 2001-2002. You will remember a flood of people coming forward after that point in time.
In those cases where the parents did report the pervert to ecclesiastical authorities, the ecclesiastical authorities would investigate the incident, and if they found it to be legitimate, they would handle it as a treatable mental disease...send the pervert to treatment. Once the pervert was declared "cured," he would be placed in another parish.
Of course, this was back in the 70s when it was believed that psychotherapy could cure anything.
And it should be noted that there were a few really notable exceptions. Out of the priests accused (4692), 149 were identified in this study as "serial" abusers -- 10 or more children.
Also of interest are the specific types of abuse reported
The point is that the definition is a whole lot more generic than what we would normally consider "sexual abuse" in casual conversation. Not that it is right. But you should keep that in mind. (Obviously the numbers above do not all add up to the number of priests and deacons accused: 4692 -- so some did more than one perverted act)
The full demographic study done on the situation is here: John Jay study
Hopefully that answers your question.