BC Fred pretty much nailed it, but I curious to know if you actually have a problem. Have you in fact experienced a change in POI or are you just worried you might because you haven't spent $10K on your rifle? If you shoot the rifle over a period of time and record the climatic conditions, particularly temperature, wind, humidity, the location, the target angle, your ammunition details and lot number etc, over time you will be able to determine if your POI is repeatable or at least what tolerances you can anticipate from it.
If you are shooting factory ammunition, you can expect changes in POI from one lot number to another.
Shifts in POI will occur seasonally if you are a cold weather shooter. Cold air is denser and this effect the bullet's flight. Cold temperature also effects the gas pressure created upon firing. Oh, by the way, Hodgdon's Extreme powder line is temperature stable at high temperature not low. At -35 expect a drop of 100 fps or so.
Although I don't seem to use them anymore, I've had pretty good luck with cheap scopes, and for years used a 3-9X40 Bushnell when I "upgraded" from a fixed K series Weaver. Once it's zeroed, a cheap scope should be able to hold it's zero. A problem might occur when you wish to engage long range targets and make precise adjustments to do so. Even some expensive scopes are not repeatable when it comes to dialing in large adjustments for elevation or wind then returning to the original zero. The test for this is called shooting a box. This is where you fire one round at a 100 yard target, adjust 6 MOA up then, using the same aiming point, fire one round, adjust 6 MOA right and fire one round, adjust 6 MOA down and fire one round, adjust 6 MOA left and fire one round. When you examine your target the first and last bullets should cut the same hole, and the other 4 holes should form the corners of a neat box. If your scope fails the test, don't be disheartened, just acknowledge that your adjustments are not repeatable so adjusting the scope for long range or for wind conditions will not produce reliable results. You'll just have to do like the rest of us and learn to hold off for the correction.
There are two stock related conditions that could lead to a shift in POI. One is that the rifle changes it's POI with changes in humidity and the other is that the POI moves as the barrel heats up. In the case of humidity, a shift of POI indicates that the stock material is not stable and changes shape as it soaks up moisture or dries out. As the stock changes shape pressure is applied to or relieved from the barrel or receiver and barrel and sight are no longer in coincidence. While a bedding job might help to address this problem, I doubt if it resolves it completely because the stock is still going to move. Barrel heating effecting the POI is a simpler fix, and is normally addressed by free floating the barrel.
One particularly troubling problem relates to a shifting POI is the rifle that shoots its first shot and inch or more away from the subsequent shots. This can be allowed for with a hunting rifle, as the rifle would simply be sighted in for the POI for the first shot, but when a guy has invested in what he has hopped would be a precision rig it's a problem. You could try various combinations of firing from a clean or fouled barrel, but it is doubtful that you will ever have complete confidence in that rifle until the barrel is changed.
Drag bags provide pretty good protection, I don't think you have an issue there. Certainly a drag bag provides better protection for your rifle than my Pelican case does for my target rifle. Not that the Pelican isn't a good robust case, but because I leave it at home because it's too damn bulky, too slow to open, and I have to remove the bolt, take off the sling, close up the stock adjustments and remove the bipod so that it will fit in the case. Mostly my ridiculously expensive rifle just bumps along beside me propped up against the front seat of the pick-up.
