"Ascending with your lungs" means varying the volume around which you are cycling your breathing. You can breathe "at the top of your lungs", which means with your lungs almost full, taking small breaths in and out, so the average volume in your lungs is very large. Or you can breathe from the "bottom of your lungs", with the lungs almost empty, cycling your breathing around a much smaller volume point. Neither is efficient breathing, and it's not what you want to do throughout your dive -- that's why you HAVE a buoyancy compensator. At the beginning of the dive, you adjust your BC to compensate for the gas you intend to use, and any compression of your exposure protection. Then, as the dive goes on and you work your way shallower, you slowly bleed gas out of the BC to retain neutral buoyancy. During this time, if you need to go up or down small amounts, you can do that with breath control -- but if you remain at a different depth, you adjust the BC, rather than continuing to breathe in an artificial manner.
For ascent, you inhale and begin to cycle around a large lung volume. As you begin to rise, you can exhale more, or exhale sharply to stop your ascent. If exhaling doesn't stop you, you vent your BC and bring your system back to neutral.
If you run out of air in the BC (and dry suit, if you are using one) before you get to your safety stop, you were underweighted. And some of us weight ourselves so that we will never have an empty BC until we get to the surface.
If, for some reason, you are unable to retain air in your BC, you should be able to swim your rig up. I don't like doing routine ascents that way, though, because if you get distracted during the ascent and forget to keep kicking, you will sink, and you must always be titrating the amount of kicking to keep your depth or your desired ascent rate. I find a neutral ascent is much more of a lazy man's ascent -- by being near neutral all the time, it's extremely easy to stop, either for decompression, or to deal with a problem or even just watch something interesting.