What’s the relationship between gain (dB) and power (watts)?

I am interested in this because I have just upgraded from one power amp to two monoblocks - the gain has not changed but the available power has gone from 200W per channel to 600W...

... and yes, the monoblocks are no louder than the stereo unit. 

Here is my attempt at understanding the issue. Typed out here more to get feedback, to get my working checked, than to tell anyone else what to think.

OK, so gain tells you how much bigger the output voltage is to the input voltage, i.e. the ratio of input to output. It think (guess) that typically power amps have an input to output voltage ratio around 100 times. 

Power considerations appear to come into play when you connect the amplifier to something, like a resistor, or a speaker. In order to hold the correct voltage, the amp has to pump out current. If it runs out of current, the voltage will decrease, the relationship between output voltage and input voltage will falter. 

With something simple like a resistor, one could easily work out how much power one needed from power = voltage squared divided by resistance. But speakers are not resistors, and music is not a steady state. I think that if one suddenly wants a speaker's driver to move, its resistance momentarily collapses, and so only an amplifier with a lot of power can hold its output voltage steady. As the speaker resistance collapses due to a rim shot or square wave, the amplifier has to dump a tonne of current into it.

In my mind big power amps sound powerful because they are capable of making the speaker follow the music. They sound more exciting, not louder. 

Wouldn't an increase in the gain setting be the equivalent of turning up the volume knob, as a default setting? (using more power by default?)

I have an amp with independently adjustable gain for each of 4 inputs. I don't use it, but my understanding is that it will simply adjust (offset) the default volume (power) level for a given input. This makes sense if you wanted to level out some inputs that might be varying in volume, but why the global gain switch? Gives you a better/preferred starting point for volume.. due to input or speaker behavior?

@rols

In my mind big power amps sound powerful because they are capable of making the speaker follow the music. They sound more exciting, not louder.

I thought that increased damping factor was part of the reason that higher-power amps have more speaker control (grip), but I'm probably missing other factors; interesting topic.

This is where my technical ignorance seems significant.  Maybe I’d need a textbook to understand this:

Of course, gain only works so long as you don't exceed the output limits, whose absolute limit in a linear amp is by the power supply rails.

This is nothing more than an unnecessarily techno-jargon laced way of saying exactly what I said: Gain is a multiplier, power is a ceiling.   

This discussion is getting out of hand as simple DC models are being compared with exotic AC ones. The math is being dumped out of a need for a model of understanding which doesn't match the challenge of the questions. The answer to all of them is... the questions are too difficult to separate into little pieces. The basics are just that, basics. The answers are beyond the understanding of most people because they encounter ideas such as statistics and psychology, which some people think do not "fit" into any discussion of music. However, they do... as well as other areas of mathematics and exotic theory, both electronic and as apparently "simple" as component design (so sorry, that is ALSO extremely math oriented). The bottom line is, most people simply cannot, will not, and don't either believe or know the answers. That means if they are right or wrong. Good luck.