When you have children and go through a divorce, you have to deal with your ex on the child custody matters. You should work as a team to make this happen. Unfortunately, not all exes can do this. When there are contentious elements to the split, things might be difficult.
One thing that shouldn't ever be negotiable in this situation is mutual respect. You and your ex must agree that you are going to respect each other. This must be universal, no matter what is going on. Neither adult should speak ill of the other one, especially when the children are around.
If you find out that your child's other parent is badmouthing you to others, even the children, you might feel angry. Take a few minutes to think about what you are going to do because just flying off the handle in a rage of anger isn't going to accomplish anything.
Ideally, your parenting plan will have terms that set the standard for respect. This gives you something to fall back on if your ex does start badmouthing you. In some cases, this type of behavior will need to be addressed in a legal manner.
Children shouldn't ever be subjected to parents speaking harshly about each other. Even though you and your ex don't get along, you are both the child's parents. This is where the respect should come from.
One option that you have is to try different forms of communication. If you and your ex can't have a civilized conversation in person, consider trying other options like emails. This might make a difference in how various aspects of the custody agreement are handled.