I seem to recall Tamiya thinner working well with Tamiya paints, but turning things like Citadel or Vallejo a bit gummy. It's been a while, though, so I may be talking out of somewhere other than my mouth...
I always use an old, bushy brush to help get the stubborn bits (tip-dry, as well as under the needle and near the nozzle inside the cup) out - just spraying mild solvent through never seems to be enough, even with a bit of intentional backflow. It's an extra step, but far from a workout. The sooner you clean, the better - dried stuff takes soaking,
then scrubbing, and comes off in flakes and chunks instead of liquid that can be drained and rinsed away.
As for the actual solvent, I've used Vallejo cleaner, isopropyl alcohol (both 70 and 90%, generic drug store fare), and glass cleaner (Windex and generic - both with and without ammonia). Balancing cost and efficacy, I generally stuck with generic, store-brand glass cleaner with ammonia. Couple of bucks at Walmart or the like for an amount that will last
years, unless you're airbrushing an insane amount. I have a (weakly) filtered spray jar for cleaning and decent ventilation, though - in tight quarters, the ammonia vapors can build up. Nothing that ever caused ME to fuss, but the onus of health and safety is on the user, in these cases.