This only happens in the kitchen. Turning on the faucet hot and cold, the pressure is fine but then drops and gets slower and slower. At its lowest, the water makes a swish, swish sound.
Difficult to say. Usually high pressure on startup and a drop off to low pressure/ low flow indicates a restriction somewhere in the plumbing system. In older homes the use of galvanized pipe is often the culprit. Galvanized steel pipe clogs with minerals and corrosion over time and will need replacing.
Is it solely this faucet having an issue? Is this faucet near a bathroom that is also having issues?
Since you say you are seeing the same effect on both hot and cold it might just be a clogged aerator. This is a mesh screen in the faucet, usually right at the outlet. Generally the tip of the faucet will just unscrew. Remove it and see if the problem goes away. If not, or if your flow problems are common in other places in the house you may need to consult with a plumber.