the simplified answer has mainly been... CP shift but that is only partially true and does not move in the right direction to what mach tuck is classically described as.
At MCD for the airfoil, flow over the wing achieves above mach 1 and as the flow at the trailing edge of the wing is usually the second Kutta condition, a stagnant point, then the flow must decelerate, depending on the velocity profile that the profile of the section results in. Shortly thereafter, or sometime quite a lot later, MDD occurs where there is a drag divergence condition that is significant in drag. Looking at the pitching moment, Cm, there is not a great change in moment, and that is curious, we are taught otherwise... but... hang on... what happens when you continue to accelerate and push the airfoil along faster? at a point at higher speeds, due to a relatively low AOA to maintain level flight conditions (or 1g... whatever floats yer boat), shocks start to develop on the underside of the wing, (surprise!!!) and that results in a collapse in CL, an increase in Cd, and, yet not much Cm change. However from the little boys room at the pointy end, stuff happens, as the loss of CL for the AOA changes the trim state, and the aircraft now has a flight path that will diverge from desired for the static trim case. take out lift and the flight path will degrade quickly. what happened to the Cm, not much, what about the stab inflow change? not much change, but the nose will drop quite happily. The concern in a severe overspeed is that the condition is divergent, lower pitch attitude results in greater shock involvement, and loss of CL component.
Whenever I have flown Mdive I do not enter with anything other than a high power setting, to allow an opportunity to decelerate by thrust reduction. In this case the Cm has not caused the nose gown pitch, the rapid loss of CL component has caused the steepening pitch attitude. Speed brakes mostly will cause a pitch up, and will alleviate the lower shock condition, except on a Learjet, where that will open up your eyes promptly. In the end, it is just a wing and a moment arm from the AC, if you have a variable incidence tailplane. High mach overspeed with a fixed horizontal stab/elevator are quite a lot more sporty.