I have been in your situation, not with the flights but for everyday life. My ex wouldn't allow my fiance of 5+ years to pick up daughter from school or her home even though it was my parenting time. Another situation was that the school called my house due to my daughter sick at school. Neither my ex nor myself was available to pick her up at the time but my fiance was available. Due to objections from the ex, my fiance couldn't pick her up at the time and she stayed at school sick.
I sought out mediation and defined my fiance as a proper person for pick up. Mediators are there to help smooth out these bumps. I recommend finding one for yourself and make a list of things to mediate at the same time, such as picking up the kids earlier than normal time if there is a special circumstance such as your flying in. Compromise and say that you will give your ex a week's notice so as to not spring it on Grandma.
Just a thought but in my papers it says I can have any compatent adult do the pick-ups. To me that is just being petty and controling. I mean what if one of your kids friends parents was doing something for all the kids and needed to pick them up? It really is about the kids and not the parents in these issues. Another example is who picks the kids up from school. I do 90 percent of the time. Why should my X care if I have to get a neighbor or my wife to do it. To me is it the same thing.
In simpler turms ask your self would your X care if anyone else did the pick-ups other then one pitucular person. If the answer is no then the X is just trying to controle things.