As a general rule, I first use the same brand of primer as the brand of cartridge case I'm using... and so, in Winchester cases, I'd use standard Winchester large rifle primers and see if they are also "loose". However, even if you are using Remington primers in Winchester cartrdige cases, the primers should NOT be "loose". It should seat snuggly, but seat all the way "down". Of course, this assumes you have cleaned the "residue" out of the primer pocket BEFORE seating a new primer... and you have seated the new, unfired primer down all the way with sufficient pressure to seat it fully, but not so much pressure as to crush the primer's anvil.
Crushing the primer's anvil due to using too much pressure to seat the new primer may cause the primer to mis-fire or "hang-fire".
You could have bought virgin (unfired) brass from reloading suppliers on the inter-net like Lock, Stock & Barrel or a gun shop in your area that sells reloading supplies and loaded them with prmier, powder and bullets yourself and saved a lot of money.
I usually buy new brass in lots of AT LEAST 100 new, unfired cartridge cases (aka "virgin brass") so that they are all from the same manufacturing "lot" and more likely to all have the same internal size, etc.
However, since these cartridges cases have only been fired ONCE as factory-loaded rounds, there is NO WAY you should be having "loose" primer pockets!!!
Like Paul S. and others have said... if the primer pockets get loose, it is usually due to your load being too "hot" (too high a chamber pressure). Loose primer pockets are a SURE sign of too high a chamber pressure unless you've reamed out the primer pockets too far yourself... and this is NOT likely since primer pocket reamers generally won't ream out a primer pocket further than it is supposed to be unless you are tilting the reamer and grinding away on the sides of the primer pocket wall.
Loose primer pockets are a result of higher than normal chamber pressure pushing back through the cartridge case. Evidence of such a happening is that you will usually see black "smudges" around the circumference of the primer sitting in the primer pocket prior to decapping the primer. If you've seen or are seeing black smudges coming from around the primer... and your primer pockets are becoming more and more loose, your loads are simply too heavy since the "smudges" are actually powder residue being forced back through the mouth of the cartridge case due to EXTREMELY high chamber pressures.
After a few shots at these high pressures, you may find that the primer pockets are being expanded by EXTREMELY HIGH CHAMBER PRESSURE. Naturally, the expanded primer pockets will NOT tightly hold the primers in place.
Other high pressure signs will be that the face of your primers are heavily flattened with little or probably NO "edge" to the primer... I.E., the metal covering the face of the primer flattens out so hard that it literally crushes itself into what would normally be a slight "opening" between the outside circumference of the primer and the inside "wall" of the primer pocket. If not such slight distance exists, this is a SURE indicator that your loads are too hot and are developing too much pressure... a DANGEROUS SITUATION!!!
With higher pressures, you may find that the tooling marks on your rifle's bolt-face are engraving themselves on the face of the primer. If experiencing extremely heavy chamber pressures, the edges around the firing pin "dent" will "pucket" and a slight "ridge" will form around the firing pin dent on the rear of the primer.
With DANGEROUSLY high chamber pressures, the firing pin "dent" will become inverted and become a "nipple" rising up from the face of the primer and extruding up into the hole in the bolt-face in which the firing pin is held. This is a sign of a load that is about to "blow" up the rifle.
For thing is FOR SURE... you NEED to get to the bottom of this situation and find out WHY your primer pockets are expanding. Obviously, high pressure is one cause... or your rifle may have a lot of wear in the bolt face or chamber and the base of your cartridge cases may be expanding due to wear. A good gunsmith can measure these things and give you an "answer".
Meanwhile, if you are exceeding the "MAXIMUM LOAD" listed in your reloading manual... you must IMMEDIATELY cease using that load and those cartridge cases since any case with an expanded primer pocket is "junk".
However, KEEP a few cases to show the gunsmith so he can see and measure them for expansion.
You have a potentially dangerous situation on your hands... make your decisions WISELY. Your eyesight, limbs and life are much more important and valuable than a few rifle cartridges!
Strength & Honor...
Ron T.