Can you complete quest lines together cross platform?

To explain, my partner is just getting into the quest line and is just nearing the War Within quest. Following that, can we complete quest lines together at the same time? When I tried to replay one of them with him, the quest point didn't advance with his.

Edit: When we were playing the quest point together, his advanced and mine did not. Is it just because it's a rerun? Or do you have to run every mission twice?

Edit 2: My partner runs on PC, I run on Xbox. My partner is usually host. We will start quest lines together (in my case I will replay a completed quest line as my active quest) and play through the first mission. After completing the mission in the objective, my mission does not progress. My partners does. Perhaps it's a rerun bug? We have not tried it with any of my new quests that he has access to yet because we're getting him caught up with the story line.