We prove results relating different partially wrapped Fukaya categories, including a K\"unneth formula, a 'stop removal' result relating partially wrapped Fukaya categories relative to different stops, and a gluing formula for wrapped Fukaya categories. We also show that the partially wrapped Fukaya category of a Weinstein manifold relative a singular isotropic stop is generated by the cocores of the critical handles and the small Lagrangian disks linking the stop. The proofs are mainly geometric, and the key underlying Floer theoretic fact is an exact triangle in the Fukaya category associated to Lagrangian surgery along a short Reeb chord at infinity.