Why are you uncomfortable with it? And, possibly they’ve only been dating 3 months. But… Doesn’t mean they haven’t known each other longer, talking longer, or friends longer then those 3 months of dating🤷 you don’t get a say on who he has around on his parenting time. Yes, you do have some say on him leaving state with child, just as he can have a say in you doing the same. But, are your feelings of being uncomfortable with him doing so valid? Or, are they not so valid? It depends on your reasoning.
Is there anything in the orders saying he can’t? If he doesn’t follow orders to see his child regularly I would say no
Girl aside, because unless she’s unsafe, it’s none of your business. Does dad treat her well and there for her like he needs to be? Is Dad trustworthy and do you communicate well with him? I personally would let my children’s fathers take them whenever he wants, because I trust him, I wouldn’t have had kids with him otherwise. If you’re personally uncomfortable with it (due to DAD, don’t involve your feelings about his gf) then don’t let them, but if you’re only problem is his gf then just let it go.
He can’t take the child x amount of miles or across state line without your permission. So if you say no then that’s the end of it. If he sees the child once a month then I’d say no. To much can happen and with once a month he more than likely doesn’t know the child very well. I’d be nervous if it was my child and the father was only around once a month. There isn’t enough knowledge of the child to be going off somewhere with. Bed routines, what to do if this or that, etc.
Me personally on what u said. I would not let them take my child across country. He sees once a month his choice? No way.
Also you could ask the child what she thinks. If child isn’t comfortable going I wouldn’t make her
Maybe the girlfriend is pushing him to see your child more. Maybe ask to meet her for coffee and get to know her more
If your uncomfortable maybe meet with them both and give your concerns. Maybe ask them to have child call you everyday and send pictures.
You can have a child custody agreement amended to include the g/f and b/f on your part cannot stay in the home while the child is there until marriage. This will stop a revolving door and stop vacations with g/f’s.
You can’t stop the father doing so on his parenting time, unless it’s cross state borders. Depending on the custody order, some you can’t take them out of a county.
I would let me kiddo go with them, if I trusted her father. It’s important for the kiddo to have a relationship with the new girlfriend, if she ends up being a long term person in the child’s life/father’s life. Trust the father enough to take care of the child, and have the best interest of the child at heart or take him to court for full custody. 
Yeah that’s a no from me dawg
If it his once a month time, there isn’t much that you can do unless you know your child won’t be safe
You need legal agreements in place ASAP or he will continue railroading you because you have nothing to back you up. Seems like he wants to “play house” with the child and new girlfriend …
If he didn’t have a woman in his life and passenger seat, would he still offer to take his child on a cross country road trip? If not, it’d be a Hell No from me. Otherwise, if he’s a consistent, trusted, co-parent, I wouldn’t have a problem with a genuine offer.
This doesn’t seem genuine, it seems theatrical (an act for the new girlfriend).
You tell him he’ll to the naw naw not happening with my kids idgaf father or not you don’t know that girl. You don’t know who she is where she’s from does she like kids or is she a girl that will only like her own kids you tell him you’re not letting y’all’s kid go with him and her till you know her better it’s not worth the risk. Too many women get jealous of the bm and kids and might hurt the child.