P!nk's day out with her baby boy was rudely interrupted ... allegedly by Dr. Luke and Kesha's legal war, and it all happened on camera.

P!nk walked out of a building in NYC Wednesday, toting her one-year-old son, Jameson. Paparazzi and fans were waiting for her, and so was a process server. You gotta see the video ... before P!nk and Jameson can make it into their waiting car ... the dude pounces with a piece of paper.

We're told the summons is related to the ongoing lawsuit between Kesha and Dr. Luke. P!nk's been a vocal supporter of Kesha ... calling the case karma for Luke, who she calls "not a good person."