I have an old house with about 4 plugs above the kitchen counter near the sink. There is electrical wire running to the plug nearest the sink and the rest of the plugs get power from that. None of the plugs are GFI. If I replace the plug closest to the sink with a GFI outlet plug, do the other plugs shut off when the GFI gets tripped (ex: water tripping the outlet)?

 

Yes, if the first receptacle is a GFCI receptacle and the other non-GFCI receptacles are connected to its load side than they will go off when the GFCI goes off.

Terminology note: Plugs are male. Receptacles are female.