Power and Sound Quality, Other Things Being Equal

Suppose that a manufacturer has several amps with the same basic construction but with different amounts of power (e.g. NAD C320BEE, C352, C372). Will the highest power amp sound better than the others at moderate volumes or only when you crank up the volume?
The higher powered amp almost always is capable of delivering the most satisfying sound, especially at moderate levels.
That would greatly depend on parts, some of a manufaturers higher power amps also emply better parts so it is hard to make a general claim that would always hold true. But as far as you asked with same general construction a higher power amp would more easily drive speakers, but not pushing any real watts it is most likely gonna sound good regardless.

I owned two amps in Classe's old CA-N01 series, the 101 and then the 301. They were apparently (according to rumor and hearsay on the internet) based on the same circuit topology, and differed primarily in power ratings -- 100 W/ch (8 Ohms) for the 101, 300 W/ch for the 301. The 101, at 1/3 the cost of the 301, sounded much faster, balanced, immediate than the 301. The 301 could produce astounding bass, but was for the most part sluggish and boring. I agree with reviews of the 301 on AudioReview.com that suggest the 301 is "polite" and "laid-back".

Classe presented the layout of the two amps in their web-avaialble product manuals for 101 and 301. The larger amp has 5 output devices per channel. I've read that having a relatively large number of devices, in which the output is delivered by multiple components, can be the cause of a sluggish, somewhat incoherent sound. Just a thought.

My BAT VK-200, which has only two amplification stages (though I don't know how many output devices) to produce its 100 W/ch, sounds marvelous -- clean, quick, clear. And my old Forte 4A at 50 W/ch was one of the best amps I've ever owned.
That said, you often get the same technology in the lowest- powered offering of the maker. The Krell 50s were thought by many to be the best of the breed despite the low wattage.

In reference to your specific question, I don't think the higher-powered amp is going sound much different than a lower powered amp when each is playing an identical signal. The high power amp will have greater reserves but the differences end there, imo.