What specific knowledge should a mechanic have in order to swap transmissions for my vehicle

I had a pretty good suspicion of the cause of the problem, but when you added, "the transmission fluid looks so good that they think it has been flushed or drained and filled recently", I'm even more confident. More than likely the wrong transmission fluid was used. Just about every vehicle since the early '90s calls for a special transmission fluid. In fact, I get the same type of shudder with my daily driver '94 Grand Voyager for this reason.

You'll understand why you're right about your multiple dandy observations when I explain how the system works. All automatic transmissions use a "torque converter" between the engine and the transmission. It looks like a giant bagel with two fans inside, and it's filled with transmission fluid. One fan is driven by the engine, and it spins the transmission fluid. That fluid spins the second fan which drives the transmission. Naturally there is slippage with that fluid coupling. That is what allows the engine to remain running, in gear, when the car is stopped at a stop sign. Chrysler developed a lock-up clutch for the '77 model year that mechanically locks those two fans together for lower engine speed and better fuel mileage. Eventually every manufacturer adopted the design, and they all use it now.

That lock-up clutch is basically two round steel plates that that get pressed together by pressurized transmission fluid, but that can't be allowed to happen at real low speeds or it would snub the engine off, just like with a manual transmission when you forget to push the clutch pedal. Typically the clutch starts to engage around 35 to 40 mph, and only in the higher gears, and only once the engine is warmed up. When everything is still cold, the transmission fluid is relatively thick and doesn't flow freely. The engineers want to keep engine speed up, which increases the transmission's pump output, to insure fluid flows to the transmission cooler. They insure that flow is established by waiting until the engine coolant reaches a certain temperature. That's why you don't notice this problem for the first 15 minutes.

By 50 -60 mph, that clutch locks up solidly, so you won't notice the shudder, but at its lowest speeds of around 35 -40 mph, partial lock-up causes the clutch to engage slowly so you don't feel a harsh thump. There's special additives in the transmission fluid that allow that partial lock-up to occur smoothly. When a perfectly fine, but wrong fluid formulation is used, that missing additive can cause the clutch discs to grab too aggressively, then engine torque tears them loose, then they try to grab again, repeatedly.

The easiest way to verify this is the cause is as soon as the shudder starts to occur, hold road speed and the accelerator pedal perfectly steady, then lightly tap the brake pedal with your left foot. That causes the computer to unlock the clutch in what it thinks is preparation for coming to a stop. The shudder will stop, then the clutch will reengage after two or three seconds, and the shudder will start again.

On a hot summer day, the fluid and coolant will get up to temperature faster, so you'll get this shudder in as little as five minutes of starting a cold engine. On a cold winter day I can make it seven miles at highway speed before my clutch engages. When it engages, you'll feel and hear engine speed slow down by about 200 rpm. It can actually feel like it's shifting up to a higher gear. If you have a tachometer in your instrument cluster, you'll see the increase in engine speed when you tap the brake pedal.

The most common transmission service is a simple drain-and-fill with a new filter. That only replaces about half of the fluid, but the new stuff has plenty of additives to keep the transmission happy. If this service, or the more-involved flush was done recently, the solution for this shudder can just be to drop the pan with the fluid in it, and reassemble, then install new fluid of the right type. No need to replace the filter again.