The original enquirer Mike P in Dallas says this only happens with hot water. Not in my experience! Mind you , maybe this is something to do with mixer taps? (Hope you folks across the pond don’t mind if I say “tap” rather than “faucet”. What do you call the combined hot & cold fittings anyway? Mixer faucets?)
I switch on hot and cold separately in the bathtub, check the temperature of the mix, and push the plunger to make the water go up through the shower hose. As I’m standing there enjoying the water, it gets hotter and hotter. To return to the original temperature, I have to open up the cold tap more and more. Personally, I like a hot shower, and the gradual increase is comfortable anyway; but my wife does object to me steaming up all the bathroom mirrors!
That might be explained by the heat in the mixer tap fitting reaching the washer by the cold water and making it expand, despite the flow of fresh cold water past the washer. I always open the hot more than the cold, so the expansion would have more effect on the cold water, increasing the temperature of the mix.
But, I’m sure that when I use the cold water alone in the kitchen, it still slows to a dribble. Again, it happens to be a mixer tap, but only the cold is flowing. Explain that?
There are a couple other possibilities as to why the hot water runs like this at the tap. Both have to do with some sort of blockage or restriction. When the tap is closed, the pressure on both sides of the blockage equalizes. When it is opened, the pressure beyond the blockage diminishes as water is released.
You can demonstrate this with a garden hose, hose-end sprayer that shuts off, and a clamp. Turn on the water and let it run through until the air blows out. Shut off the sprayer (not the faucet), then partially clamp the hose about halfway down its length. Turn on the sprayer and the pressure will slowly diminish. Turn off the sprayer and pressure will build up again.
Sediment and deposits in the hot water heater are causing blockage of the dip tube or other supply lines.
If there is blockage somewhere on the cold water side, then a similar effect will occur. If it has been gradually occurring, then the typical causes are corrosion, sediment, and deposits inside the pipes.