I think it's obvious from what we know of Otsutsuki clan culture and Urashiki's personality that he's not going to grieve for Momoshiki or anything. I doubt he's going to carelessly act on earth again either after he got his ass whooped and his seemingly higher caste comrades were killed; he'll probably peace out to the Otsutsuki homeworld and report what happened to the clan.
I don't think Urashiki and Momoshiki had a particularly bad relationship, though. Although he'd logically have to be ancient to have known Kaguya, it's made clear in the Boruto novelization that he's considered a child by the clan, with Kinshiki as his "parent." It's not weird that Urashiki, who is presumably an adult, would look down on him. I think Urashiki only deferred to him in the first place due to his laid back personality, or perhaps because the Otsutsuki have a caste system or something, which would also make a lot of sense.