I don't wash after every session. I wash everything once a day, bottles, connectors, everything. Bottle brush for the bottles. For the pump connectors, Dr. Brown's makes some very small brushes that fit into those holes and cleans them out. I also use a nipple brush for the teats my baby uses, and it cleans the connectors pretty well too.
Dishwasher just doesn't work for us. I think it's the hard water here.
I use blue Dawn. For that residue, a soaking in 50:50 water:vinegar cuts it away easily.