A Lee neck sizer collet die would be very useful in this situation. It can tighten and standardize neck and tension. Something is wrong in your setup or adjustments. If the bullet falls in, you are not correctly sized. Die/shell holder adjustments may be off. My approach when I have a problem is to do each step one at a time to isolate where it is going wrong. You need to trouble shoot before using a progressive press. Turn off the indexing/advancement until fixed. Have calipers, case gauge and bullets on hand to test fit.
1. Deprime. Bullet will fall into case.
2. Size: Bullet should now not fall into or be easily pushed into case. If it does, STOP. You are not sizing correctly due to any number of factors. A full length resizing die with the neck expander ball in it, all properly adjusted, will NOT DO THIS. This is where you want to test with a bullet, will it drop through? Put the "sized" case in a case gauge for the correct caliber. Get out your caliber and measure inside and outside of the neck versus SAAMI specs.
3. If all that checks out, Are you using a "powder through" die that includes any flaring or expansion? Charge a case in it, and measure again.
I am thinking your sizing die setup and adjustment may be off. I recommend that you download the instructions from the die maker and re-set-up, from scratch following instructions painstakingly.
The case mouth in contact with your drop tube may work just fine. I use a powder funnel a lot and don't need to expand or flare.
I had this situation happen once. I had primed un sized case and thought they were sized. I had to disassemble some rounds and resize them.