In any amplifier, if a switching supply and a linear supply both have exactly the same output voltage under no-signal conditions, the linear supply will sag to a lower voltage than the switching supply under heavy load. How much the linear supply will sag has to do with the amount of capacitance on its output. So in this instance, the linear supply is at a disadvantage and will supply fewer watts under sustained heavy output than the switching supply.
If the linear supply has a higher output voltage than the switching supply but sags to the same voltage as the switching supply under sustained heavy load conditions, then the linear supply will have the ability to deliver higher output wattage for a short period (in other words transient output), and will have the same ability to output sustained wattage as the switching supply. This is typically how a linear supply is implemented (at least in our amplifiers) so that there is transient ability above what a switching supply can offer.
Looking at @amirm amp tests it looks like most of the class D amps do not have a lot of "headroom" for transients. However I watched the QSC video linked a couple posts above this and it sounds like they have plenty of headroom on their new class D amps. I would love to see one of these amps tested. The DPA 4.3 is less than $2,000 for 4 Channels of 625 watts continuous power with almost 3 dB headroom. Looking at the specs it looks like the high SINAD fans might be disappointed but power / dynamics fan may be happy. I know I should buy one and send it in but I am not going to unfortunately.