My boyfriend lives with us. Finances, kids, household responsibilities, etc., are shared.
My husband moved out for a year, not with someone else, but on his own. Our kids went back and forth. He and I dated while working through some of our own shit. But my bf stayed here with me and the kids.
I generally see this as "neither relationship has priority over the other," but that doesn't mean that they are the same.
One detail that helped with us for clearing up some of the confusion was this.
Each parent has a responsibility to maintain the relationship they have with each child. That means they need to spend quality time with the kids. The kids being with one parent doesn't negate that the other parent needs to spend time with them.
So Maca's "free time" without kids doesn't increase just because I'm a stay-at-home mom. He has X number of hours free after work, and until the kids are in bed, those hours belong to them. On weekends, at least one day's-worth belongs to the kids. If you took the 10 hours a day I spend with the kids and expected that he "owed" the kids the same, he would have no free time at all. We don't do that, but we do make sure other lovers know that he loves his children, and his devotion to time with them is not relevant to my availability to them.
Much of that time is spent together as a family, but that family time is for the kids, it's not something to be countered between lovers. So if a girlfriend feels like she isn't getting enough time with him, the "competition" of time isn't with me, it's with the kids, and her options are to be the kind of person who can be a good addition to the family for the kids (in whatever capacity) so she can spend time with him with the kids (and potentially GG and me), or suffer the lack of time, because he's not giving up his time with the kids for another lover.
The same is true for me. But GG did want that time and he does prioritize the kids, and has chosen to give up freedom in his own apartment so he can be here with me with the kids. So he sees me daily, but he shares my time with the kids.
Any lover could choose to make life changes for that to be possible, if they wanted. None are less important than me just because I have the marriage certificate.
Likewise, all that can be said for the home maintenance, daily chores, etc. We've been building a new deck and painting the outside of the house, creating a garden, working on the acre of land we have. Others can join in, but we aren't going to neglect our responsibilities to those things for a date. On the other hand, if a lover chose to help, we'd finish faster, which would mean we'd have more time to go out on a date.
It's all about understanding that not being hierarchical doesn't mean throwing all previously agreed-to responsibilities (like bills, home maintenance, chores, kids, etc.) to the wind.
When Maca moved out, he was still responsible for his financial part of our family, because I am still home with our homeschooled kids, and it was still the choice of all three of us to maintain that for the kids. So he paid his portion of expenses here and another apartment. If he couldn't afford that, he would have had to reconsider his options, because he won't neglect the kids' best interests.
Likewise, I am in school. But if my schooling takes away from being able to keep up with the kids, I have to skip a semester or reduce my class load. I can't do full time, for example. I don't have the time. I have responsibilities I've already committed to with the kids, and I have to work around those.
Dating is the same. It has to work within the parameters of existing responsibilities. But to be honest, I have not struggled with finding lovers who are willing and able to work within those parameters. And the only reason Maca and E couldn't keep dating is because she moved too far away for work, which was her obligation and choice. If finances improved to where he could afford to travel, I wouldn't have any issue with him traveling to see her. And if she called to say she wanted to come up with her kid to stay for a few months, that would be ok too.