What happens to the disolveable plastic that on dishwasher tablets when they go down our drains...?

In theory, that "plastic" is a gel, not real plastic.  Much like the gel on the outside of pills.  This gelatin dissolves (breaks down) when it come in contact with water.  The residue is filtered out at the waste treatment plant (or if on a septic system, is removed when the tank is pumped).  In theory, it is harmless.

I don't use tablets but they go into the sewer system and eventually back into our drinking water.........thirsty?