This is one that has as many answers as there are stars in the sky, and takes a world of negotiation sometimes to balance different people’s comfort levels about whether to introduce partners to kids sooner, later, as “friends” first, or something else. Other times, it’s remarkably simple - all parties agree on the speed to introduce new partners to kids, whether it be mutually quick so kids can have playdates while parents hang out or mutually slow because everyone involved introduces kids to partners only once they’re sure the partners will be sticking around, to the best of their predictive ability.
When it’s hard, the best advice I’ve ever been given is to explain why you each chose the position or strategy you did about this issue. My personal “slow and steady,” maybe introduce people as “mommy’s friends” a couple months in if I think they will stay mommy’s friends if the romance doesn’t work out, and wait at least six months to explain that this is mommy’s partner or boyfriend and to be kissy or traditionally PDA-ish in front of the kids as opposed to just hugs or sharing a couch as I would with any friend, and an identical timetable at a minimum to introduce that partner’s children to mine if applicable, is a strategy I’ve adopted because I don’t want to traipse adults who could be important or kids who could become close friends in and out of my children’s lives. It’s sweet to have them ask for a playdate with a partner’s kids once that’s reasonable and available, but the same question after a contentious breakup, when they’d met a couple times, and now I have to explain to a five year old that their new friend’s parent doesn’t want to spend time with mommy so we aren’t going to be seeing them for a while is something I want to avoid for both my kids.
(Drawing by the older of my kids when he was 5)
I have friends (including an ex-partner) who adopt an opposite position, that it’s better to introduce children to partners, especially partners with children and their children, early and casually (like in group settings), because then it won’t be such a jarring transition if partners become “just” friends at some stage and it (from their point of view) encourages building community. I think this works better with older than younger kids; older kids generally have a better understanding of “we aren’t doing this for a while but we’ll be seeing those friends again in groups in a month or two but probably not alone so much because mom and <person> aren’t dating anymore” than little kids, and knowing myself, I generally need a little space before I’m back into full social mode with exes, even if we land at good friends.
I also have friends and know people who have positions in the middle, or positions that vary based on availability of sitters and distance of a particular partner. If someone is far enough away you go visit for a whole weekend or longer, you may either wait more time to introduce your children or bring them along on visits, depending on availability of close friends or relatives to watch kids for a couple nights in a row and whether it’s “plane and every four to six months” distance or “drive and monthly or every couple weeks” distance. The latter are much more likely to involve earlier introductions, whether for regular visits or for combining holidays.
That’s another bit of this conversation - are you going to include each other and each others’ children in your holiday celebrations? To some extent this is simpler if one partner doesn’t have local family-of-birth so being each others’ family-of-choice is an obvious shift in a given year, but the same concerns about “am I setting the right or wrong expectations that this may be a tradition for my kids?” may be a concern for one or both partners who is a parent, and talking it out to make sure it isn’t just an NRE-based choice or that there’s balance struck between positive interactions for the kids and positive interactions for parents. The larger number of adults who care for them gained by kids with a polycule who care about them (both in parents’ partners and in extra “grandparents” they might gain down the road if some of those partners are relationship-escalator partners who go high up it and have accepting parents who are glad to have more kids in their lives to love and grandkid-spoil) are a great benefit to kids, as are more kids to be friends, potential sibling-like relationships, or cousin-like relationships depending on relationship structures. We've already discussed the only real possible negative - attachments that disappear on kids can be a little unstable and it's up to the adults to negotiate that, but it's easy to balance that with timing, with expectation setting, and maybe (for people like me who'd rather have a couple months of space after a breakup) with sucking it up and having a couple spaced out playdates to slowly phase out time kids spend with kids of former partners.