caulking

Which kind of caulk?  Vinyl, acrylic, butyl rubber, latex, silicone or some variant?

Pulling. Why would you push and risk gobbing up the caulk as it's being pushed back into the opening?

If you mean the direction of travel of the tip, you push caulk so it goes into the crack. If you are pulling it, the caulk just lays on the top. The painters showed my wife how to do this and she is a lot better than me. We also have a real caulking gun, not one of those $1.99 stamped steel things. This has a cast, swiveling barrel that holds the caulk tube, a pressure release that works and a better trigger mechanism that allows more precise metering of the caulk. Getting the tip cut to the right angle and size is very important. If you do this right, you get a very consistent cove shape caulk line with mo clean up necessary and the caulk is pushed into the gap so it won't just fall out or leak.

I've never had a problem with caulk entering/filling the crack while pulling. With the tapered cut and angle of the tip, it lays a perfect bead as you leave it in it's trail.