V=IR (Current X Resistance) - so as resistance drops, current needs to rise to maintina voltage
Speakers / amps are voltage devices - that is to say the Sound Pressure Level is proportional to the voltage.
SPL specs are commonly measured at 2.83V at a distance of 1m from the speaker
for 8 ohm speakers that would be 2.83/8= 0.35375 Amps or/and W=VA so Power is 0.35375 * 2.83 = 1 Watt
(for my speakers that would be 86db/wm)
Now what if that speaker is a 4 ohm speaker - 2.83/4 = 0.7075 Amps, W= 0.7075*2.83 = 2 Watts
Same voltage - requires twice the current and twice the power output
At some point the power supply reaches its limit in terms of current (usually before the power limits of the power amp circuits are reached!)
for 40V output 8ohm ... 40/8=5 amps W=5x40 = 200W
for 40V output 4 ohm... 40/4=10 amps W=10x40 = 400W
Basically - with lower impedance speakers, the receiver had to provide double the current to keep the voltage/loudness at the same level (which results in double the power - for the same SPL)
More current means more heat ... and typical AVR's will max out at around 7amps
Once you hit the current limit of the power supply, the voltage starts to sag - you start to get various interesting audible effects... compression, congestion of soundstage etc...
Also, if the power supply is not adequately protected, you can get a literal melt down, release the "magic smoke".
So, to protect the system, designers put nanny circuits, and/or hard switches for lower impedance speakers.
Typically the "switch" simply constrains the power supply to a lower maximum voltage... which results in lower current output (typically gauged to be within the power supplies long term capability - ie: it won't overheat / melt down / fail ) - which in turn results in lower max power output. The Nanny circuits similarly just drop the supply voltage to the power amp circuits.
Top of the Line AVR's of 15 years ago, frequently weighed 30kg - this was mostly due to the massive transformer, giving them higher power, but more importantly substantially more current - meaning they worked better than most current AVR's with low impedance speakers. - my old TX-SR876 could handle my 1.6 ohm speakers well, my current Integra DRX 3.4 can't (the former weighed 30kg, the latter 10kg... both are class AB difference is almost entirely in the transformer)
In terms of whether the power output rises or falls - is a consequence of all the above...
Pretty much all current amps have more power at 4 ohm than at 2ohm (cos for the same voltage output, it would provide double the power, if not current constrained) - but then look at amps measured at 2ohm
The Quad 606 is a 135W amp @ 8ohm, rises substantially into 4 ohms - 180W (notice it does not double - it has reached its current limit...) - and having reached its current limit, the max voltage it can support (cleanly!) then drops off dramatically as impedance drops further - for 2 ohm we are down at just under 90W.
Inveresely - that Quad 606 - can drive a 2ohm speaker to the equivalent loudness of a 22.5W amp into an 8 ohm equivalent speaker (it's max voltage/loudness into 2 ohm is severely constrained).
The nice thing about the Quad is that although constrained, it still keeps going, remains stable, and sounds good (as long as you are not asking massive SPL's out of it!)
Most amps, and most AVR's simply start to sound like rubbish into such 2 ohm or sub 2 ohm loads - and if not adequately protected, may well self destruct.