Will the flow rate of water slow if the diameter of the pipe it runs through is decreased?
If there were no frictional losses, no, the velocity would increase to maintain the same flow rate which is velocity times area.
However
In real life as the velocity increases the losses of total energy due to friction (viscosity) increase and the flow will slow down for a given total head change. In other words partially clogged pipes do not work as well as clean ones :)