Does probiotics bacteria from yogurt live in the mouth too? Does it populate the mouth/throat or just the gut?
of course they do... the question is for how long.
Probiotics will tend only to affect the microbiome in the digestive tract for a while.  For yogurt supplemented with probiotics subjects microbiomes drifted back within 48 hours.  Depending on your diet the individuals can hang out for a while or persist.  Children, who tend to drink more dairy have a more persistent Lactobacillus signature.