Layer them

One approach is to lay the t-shirts flat one by one until you have a flat 'stack'. Then pick the entire stack up by the middle, creating one large, round, fold. Put the bundle in the case.

This is the approach I use to help stop them getting wrinkled. Which is what happens when they move around a lot.

I combine this with the other packing techniques described in the other answers to pack additional materials around them and use a case of the right size to hold them tightly without unused space.

I just use a suitcase which is just big enough for my needs, in that way there is no space for clothes to unfold.

This is helped by using the straps in the case or a packing cube with other items on top when the cleans get used and moved to a 'dirties' bag. The items which were in the dirties space will then move to the cube over the clean shirts (and other clothes.)

To build upon Kate's answer, one can use an actual box that fits the t-shirts' dimensions, e.g. in my case:

Even though the box was advertised as a shoe box, that did the trick (dimensions in inches because in the United States):

They can be stacked too in case one has a large suitcase or prefer to directly place them in the trunk:

Best way of keeping things folded is to put in some other material as padding, assuming that there will be free space left in the suitcase. Kind of similar to trying to figure out how to pack delicate items in a suitcase such that they won't move around too much, risking that they will break.

It may be best to try and consolidate suitcases if you are packing more than one, to help ensure that things will be more tightly packed and reduce the movement/shifting of items within the suitcase. If your suitcase has any straps or smaller compartments, you could try making use of those as well.

Although I should say that items are less likely to shift if you're driving on the highway for long periods of time, as opposed to an aircraft where there may be more movement, especially during taxi/takeoff/baggage handling if you are checking it in.

Empty space makes things more likely to move and therefore unfold. You can add light padding to occupy the space around. If that is feasible, it will limit unfolding.

The other thing to use is compression bags. There are essentially a ziplock bag with a valve. Items are folded into them and all the air taken out. Some people use an actual ziplock bag and a straw to do this on the cheap but you probably won't get as much compression. Once air is outside of the bag and your clothes are tight, they are unlikely to unfold, even if they move around.

Some people use an interlocking folding method to keep things together and more compact but this leads to a hassle when trying to get items out and then more work putting things back in. It reduces chances of unfolding but does not prevent it like compression bags.

how to reduce the likelihood that they unfold?

By not giving them the option to unfold.

And my preferred way of doing this is by not folding them in the first place, but instead rolling them (and securing them individually with elastic/rubber/gum(1) bands)

Rolling also reduces the volume clothes takes up, so you can fit a greater mass in the same volume.

(1) Depending on your preferred dialect of English.

I like to use packing cubes. These don't have any kind of vacuum seal, but they do kind of squish their contents a little and let you pack more than you otherwise would. If you put all your shirts into a cube that is a tight fit for them, and zip it shut, they will not move around within that cube.

(Despite the name, they are not cubes, proportioned more like a book.)

You can use one for tshirts, one for pants, one for bathroom things and so on. This makes unpacking at the other end simple and I never worry that the dressers or closets might be dusty or the like. I just put the cube in the drawer or on the shelf and then open it.