Personally, I wouldn't let my kid meet a partner for months and months. Recently, my son had to meet one of them because he left something at my house, but he was introduced as a friend and that's all that partner will ever be in my son's life anyway.
I think I need to clarify what I mean by all this a bit more. Of course, everyone has boundaries in their relationships and things crop up that you didn't predict were going to be an issue but turn out to be, and things need to be re negotiated. That's all fine. It's important to have contemporaneous communication with your partner(s) and sometimes your metamours too. However, the idea of having to police their other relationships because your partner(s) are unable to maintain those boundaries is undesirable at best. Maybe once or twice something will crop up that wasn't handled as well as it could be, but especially when it comes to children (I've expressed how I handle relationships in relation to my son), even once where he didn't step up and assert the agreements that we have made would be too much for me. But the focus of my "blame", for want of a better word, would be on him. I wouldn't think that it means regularly and routinely having to manage his relationships in that fashion was desirable and if it became so I had to do that in order to keep our relationship healthy, it just wouldn't work for me. I need to be able to trust that my partner(s) can maintain our relationship whilst forming and maintaining others without me having to oversee it.