I feel badly for the 10 year old. I hope her Mom sticks with the new date. Have to agree with the posters that said go for her sake. Kids need all the love and support from family that they can get these days.
In our side of the family that is NOT a problem... we simply discuss a date beforehand if it's important to us that most everyone can make it. Instead of changing many times as you find out it's a problem, I would suggest contacting the main people involved to see if a particular date would work for them, or if there is a conflict. Then set a date based on your findings. When I get an invitation I put it on the calendar and work around it. If it's changed several times, not only is my calendar a mess but also there is less chance I will be able to come.