I've seen this condition many times and usually it's not the gun. First, make sure all your scope and mount screws are tight. Then check the front and rear stock screws for tightness. Give the bore and chamber a good cleaning.
A trick I learned years ago is a poor man's barrel free float. Get a flat washer with a hole large enough for the front stock screw. Put the washer in a vise and squeeze it until it becomes slightly arched. Take the barreled action out and insert the arched washer between the barrel and the stock with the front screw going through the washer. Reassemble and tighten the screws. You should now be able to slip a dollar bill between the barrel and forend. I've done this to many 700's and it always worked. If you don't like it, it's very easy to reverse. This is good for tightening groups but probably won't fix your flyer problem.
Likely what's happening is the cartridge in the chamber is getting hot from the previous shots. When powder gets hot, it changes characteristics and increases the burn rate. In turn, the chamber pressure and velocity also increase. Of course this will cause accuracy problems. Nothing you can do to the gun will correct this problem.
If you reload, use a powder such as Varget or H322 that is rated for temperature extremes. I would recommend waiting 5 minutes between shots when going for groups. Chamber the cartridge just before shooting. This technique will do more for flyers than any thing else (assuming all screws are tight).
Another thing that will cause a flyer is the parallax adjustment on the scope. If your scope has an AO adjustment, make sure it is set for the correct distance. If not, your eye must track perfect through the scope with each shot or your group will spread because of parallax.
