Powered subwoofers running into power amp?

You will overload the input capacity with a high level amplified signal. The resulting sound might be louder but would likely be badly distorted.

Actually...plenty of powered bass bins (Mackies for example) have a "pass through" feature that will work exactly as you described. I've sent stereo signals to a powered sub equipped with SIGNAL outs from it to pro powered main speakers, and it works perfectly. It's how the pro sound world works (mostly). You can take these same outs to any pro power amp hooked up to passive speakers (which I've also done about 462 times), and your level will be cleanly controlled by the mixer, unless you have no clue what you're doing and crank everything to the point that it explodes and causes your girlfriend to dump you for an ear specialist. The Mackie site has all this spelled out.

The signal from the bass subs is not amplified. It is just passed from input to output by straight wire. It can be sent to power amps directly.

The only issue is loading of your mixer output with the additional input impedances (resistive and capacitive loading) of both the bass subs and the power amp, plus the capacitance loading of the interconnects. The bass sub input impedance and the power amp input impedance will appear in parallel with the ouput of the mixer.

So, for example, if the power amp input is 50K, 100pf and the bass subs are 50K each, 100 pF, then the mixer output will see 16.7K and 300 pF. If it can drive this load, then no problem.