As long as your molds are for bullets with the standard lube groves then yes you can run then through your luber/sizer. Lee should be able to provide you with the proper nose punch. 
Some of the molds that lee offers are specific to tumble lube and I have never used any of those and don't know if they will hold much lube from a luber/sizer. 
Another option is to pan lube your bullets and then shoot them as cast. If your using low pressure loads such as black powder this will not cause any problems unless your bullets are grossly oversize. 
One last option that I can think of is to pan lube and then size them with the Lee push through sizing dies. The up side to using the push through die is that it pushes on the base of the bullet so any bullet of the same size can be sized regardless of the nose profile.
Sam

Cool Cat, I didn't know that, never really looked at the prints for the TL bullets.
Sam

I've used my Lyman lubrisizer to lube Lee tumble lube bullets and they worked just fine. If the bullets need to be sized down much it will reduce the amount of lube, but still have more than liquid alox provides.

Welcome to the wonderful world of bullet casting. Lee molds are a great way to get started. After you get hooked you'll find that a good custom mold is a lot more satisfying and well worth the extra money. If your old RCBS lubri-sizer is old enough to use the ratchet handle, hang onto it. I like them a lot better than the new style. A friend bought one of the new ones and after some use, converted it to the old style.

JS

JS47 wrote on Nov 5^th, 2017 at 10:34pm:

Lube under the bottom usually means the sizing die has too much clearance for the bullet. I get that when I'm trying to just lube and not size the bullet. Even .001" oversized die will allow lube to get under the bullet. I just wipe it off and every so often clean the die off the top.

Epoxy in the top punch is a great idea. I use epoxy putty, it's easier to handle. 
Try adjusting the stop of the luber slightly up & down for the lubing problem.