I don't feed probiotics all the time, I've never felt that a healthy dog on a good diet has any need of that, but I do always have some in the cupboard and add them to the diet when they are needed - I think they are beneficial whenever the dog has a digestive upset or diarrhoea (such as after picking up & eating something nasty on a walk), after antibiotic treatment (since antibiotics kill good as well as bad bacteria), and at times of emotional stress (I believe that can also upset gut bacteria).
When I do give probiotics I choose one that includes prebiotics in the formulation; they are more expensive but since I don't use them all the time I'm not too worried about that. The one I have in stock at the moment is Yumpro.
A rescue dog such as those you mention would be another matter of course, possibly needing long-term help, and I'm afraid I really don't know how that should be approached. Hopefully someone else will know more.