Bathtub starts draining normally but then slows down
Sounds like the kitchen sink line has a 'fat berg', albeit a mini one. If the water is backing up along the sink line it could easily affect other connected pipework i.e. the base of the vent stack etc. I'd be tempted to start at the kitchen sink anyway as it's very often the cause of problems due to the gunk which goes down it. Hot greasy stuff goes into the sink, but as soon as it hits the cold pipes a few metres away it solidifies and builds up; often where the pipes head outside.

Grab yourself a spiral drain cleaner and remove the trap under the sink and gently clear out the run away from the sink. Finish off with a chemical cleaner if you like and plenty of hot water to flush away any remaining debris.