I always distract them with something else that is of their interest to get them from forgetting why they were crying about. At a store they hold something I immediately get their attention on some other toy before they get too attached with what they are holding.
At home when they can't share toys with siblings I let them spend 1 0n 1 time with me doing my stuff together. Usually the other kid will become jealous and want in on it too and they both play together again.
Certainly not full proof but most of the time it works.