Op as someone a bit further down this road, I can only give you advice based on what worked for me.
When exh and I moved out of the marital home (we had been separated a while) he moved into a rental. 2 weeks later, he was introducing someone to the kids. 10 weeks after that he left the rental and moved in with her. 4 weeks later he was engaged. He got no reaction from me.
6 months later they split. My dd is older was 14 at the time refused to meet another girlfriend. But in the 2.6 years since he has had 3 more girlfriend and engaged another once. Which has just ended.
Exh did the same with ds. Asking him, if I mentioned anything. I hadn't. I once asked ds if he had a nice time with his dad and he said 'dad says I don't have to tell you anything if I am not comfortable'.
I didn't react and said 'well that depends. If someone is hurting you, you should tell me. But do you mean you have seems dad girlfriend and don't want to talk about it?' And he said yes. I told him I was just asking if he had a nice weekend and I didn't need details. But also told him, he can talk to me about anything and it would stay between us 2. As far as exh knew, there was no reaction.
Eventually, ds started mentioning her in passing and I would just listen and never ask questions.
I had the first girlfriend stalking me on Instagram. She still does on occasion now. Still no reaction I just block and move on.
He has just split up with one and started seeing another. During this time he has also moved 6 times.
He has always prodded and prodded me until I reacted. This time I haven't and it infuriates him. But the kids see him for what he is and just aren't bothered by him anymore.
I gave the kids stability. I haven't had a string of boyfriends ( I do, now, have a dp, who I took it slow with), or constantly moved house because debt collectors caught up with me or dropped them short notice or not turned up to things. As a result both kids don't really see him anymore. But they are more happy, confident and settled for it. My lack of reaction ensured that they felt safe and they can tell me anything. They didn't see me upset or angry over him. Or have to worry about upsetting me. And all three of us are better for it.
No reaction to things you can not do anything about, is the best way.