OK let's break this down.

Distilled water will not in fact do any of this at a tiny l level. For one, again, the difference between distilled water and tap water in terms of osmolarity is actually pretty much insignificant.

For two, your mouth, and esophagus have mucus linings making this entirely moot as far as water is concerned.

For three, distilled water is far and away the least stressful thing your body ever deals with, eating something even moderately salty is far far more "dangerous" to your cells than drinking distilled water. Ever eat a potato chip?

Third, your cells actively resist osmotic pressure changes as necessary. You have to go pretty hard and actually change the salinity of your blood to have any noticeable effects.

Fourth, your cells are actively being replenished with good salinity blood.

Fifth, we are very insensitive to salinity changes in our blood. Blood salinity shifts quite a bit and regularly throughout the day.

Sixth, a benefit of being a multicellular organism is that we don't care much about any individual cell. Your mouth cells die and are replenished nearly the fastest of any set of tissue in the body. Your mouth in general is very much built to handle all sorts of harsh conditions. Chemically distilled water is nothing. I guarantee you 100% that if you drank distilled water every day for your entire life, and made sure to eat a normal diet which was not extremely low in salts, you would be fine.

Let's discuss non multicellular organisms for a moment as well: the salinity of fresh water is very low. That's the point of distinguishing it. Plenty of organisms live just fine in it. If you drank those organisms, many of them could also get you sick, as in live inside you, a pool of water with much higher salinity. Cells have ways to maintain homeostasis and respond to their environment. That's literally what they do.

No I don't think there is research on this specific subject. Are you starting to see why there isn't much worry on microdosing distilled water?