One of the best example is Ursula Von der Leyen. She was sent from Berlin to EU by Angela Merkle because when she was the defence minister of Germany under Merkle’s government there was a huge scandal of offering contracts (https://www.euractiv.com/news/von-der-leyen-admits-mistakes-...)

She was sent to EU so she can do less harm in the future, but the irony is that she became the head of EU.

MEPs in many countries do not actually have immunity.

However, the bigger issue is that european parliament is treated in many countries as essentially dumping ground between national elections.