You can use the change of circumstances as the reason to file for modifying custody. I suggest that you would need to probably discuss with her about a possible trial change prior to going back to court.
We are sort of in the same boat with my oldest stepson who is soon to be 13 but the advantage to this is that their mother does live within 30 min drive and that they have had shared custody with equal time since their separation.
If it’s unlikely that she would agree to this you could probably file for modification for joint physical custody. There are a lot more courts ruling in favor of shared custody than in years passed, and to be honest, the courts are more likely to rule for joint custody than a complete change in primary custody. Don’t discredit the neglect. If there is documentation in the past keep up with it. The courts can look at past parenting practices to see how a parent will likely behave in the future. If they were being neglected, it’s rare that that just stops. With that many people and children involved, it’s unlikely that they are receiving the attention and devotion that they could get from you.
If you do file, I would do it when the children are with you. The first day that you have them, ask them if they would want this change, and contact an attorney. Document everything.
If you have an agreement spelling out your visitations then do NOT change it. It’s possible that there is something planned for the month of July but if your ex is anything like my husband’s it’s more likely that there is a desire to split up your visitation schedule so that she has more influence over the children. This way you don’t have them for a solid period of time and there is less chance for them to get used to a schedule with you…