What exactly constitutes a "change of material circumstances" regarding child custody in a settled divorce case?

Your question did not address if the children live with you or not, but for purposes of this answer I will assume they do.

Your getting married should have no effect on the custody unless there is some aspect of your new spouse that would cause a Court concern to have the children living under the same roof as the spouse (an extreme example would be a history of felony convictions for child abuse or molestation).

Could your ex bring a suit for modification of custody? Yes, he could file one. That does not mean he would prevail of course.

In relocation cases the primary consideration is the best interest of the child test, all other rights are secondary. Any determination is made on a case by case basis and there is no presumption that a relocating parent will always lose custody just as there is no presumption in favor of relocation. Before a custody award may be modified there must be a finding of a new material change in circumstances affecting the child.

There are MANY factors which have an impact on the children that will be considered such as: a child's relationship with a non-custodial parent, the child's ties to local schools and friends, the child's age, the stress and instability of relocation and the corresponding benefits of consistency and stablity for the child, interests of the entire binuclear family (consisting of one household headed by the custodial parent and another household headed by the non-custodial parent, the child being a part of both), the custodial parent's reason for relocating, the dynamics of the custodial parent's new family unit, and any other relevant factors that may need to be taken into consideration.

To authorize a change of custody, it must be shown that there has been a change of condition which substantially affects the interest and welfare of the child. The changed condition must be material ("substantial") and the changed condition must be shown with fresh evidence that the change has occurred since the last prior award of custody.

So....the children's father could likely bring an action but whether or not it would result in a modification of custody would hinge on various factors. Furthermore, it would be difficult to predict how the Court would weigh and rule on those factors with any certainty without knowing much more about the case.