De-escalate the level of urgency in every email
When you get an email that is angry or accusatory it's not enough to resist taking the bait, to wait and reply with a cool head to ensure you don’t engage in an argument. You need to actively de-escalate the level of anger - choose words and tone that brings everything back to normal.
It’s difficult and it requires a lot of restraint and maturity. You’ll have to find a way to elevate yourself enough so you can resist lowering yourself to their level. If the relationship is going to remain productive, one of you is going to have to be the bigger person, and it’s not going to be them.