Do oral probiotics really have an effect on other parts of the body (like the skin, lungs, etc.) and, if yes, how?

I often hear that probiotics can help with urinary tract infections and the vaginal microbiome. I also recently heard that they can help with/protect you from respiratory infections like COVID-19.

I am having a hard time imagining how this works. Am I missing something? Are these articles referring to oral gut probiotics (like yogurt, kefir, sauerkraut, probiotic pills, etc.)? Or are they referring to a completely different kind of probiotics applied directly to the site? How could gut bacteria have an impact on the urinary tract, skin, vagina, lungs, etc? Do the gut bacteria stay in the gut or can they migrate to parts of the body (like the skin or urinary tract)? And do they have some kind of effect while in the gut that can somehow impact areas far away (like the respiratory system)? How does this work?

I am aware of the gut-brain connection claims, which make more sense to me, but some of these other claims seem strange to me.