This won't change if you do it base, first. The reason the checks are loose is the gas check stem is too small. Lapping out the mold is the only definitive cure... unless you want to size your bullet a mic or so smaller.I have had checks pull back through the die, tapping them back on, on the bench top just leaves them loose. I don't like loose checks. The check has more friction then the boolit in the die.
This might be a case of a burred sizing die. I little lapping of the opening might help?I have had some checks fold over themselves by catching the edge as they go into the die body.
If your bullets are too small to get good friction in the sizing die, it might be possible that the checks don't get seated quite fully. But nose-first is my preferred method. Personally, I'll preferentially work out the kinks to make nose-first work. Base first, the check will potentially swage itself to the base of the bullet, rather than perpendicular/flat. I dunno about your guy's bullets, but the base of my boolits have a little nub where the sprue cutoff is. I'd call the Lee pusher surface much more flat and perpendicular than the base of my boolits.