I cannot see how one can completely automate batch processing in this scenario.
The issue is not only fluctuating flash output but distance to subject as well.
The only way that I know to do this is manual comparison.
>Is that because the ISO, aperture etc. is the same in all the pictures
Yes. That is exactly what "Match exposures" does. It looks at ISO, aperture, and shutter and corrects to get to the same EV. If all your images were shot at the same settings, they can't be corrected this way. This is not the way to do consistent flash exposure BTW. You should be correcting either flash output or aperture based on distance (or ISO if you don't want to change aperture or flash intensity) as the intensity of light on your subject decreases inverse square law with distance. Most camera systems can do this automatically based on focus distance and some through the lens flash metering but it is possible to do this manually too. If your flash is off camera (e.g. shooting through an umbrella or bouncing on a ceiling or somesuch) you often don't need to do this and you can stay at one exposure since the distance of the flash to your subject generally doesn't change and so amount of light on the subject doesn't vary.
> I need a way to correct for the fluctuating light output of the flash.
Can't be done easily in Lightroom. The only way is manually correcting visually. It is almost impossible to automate something like this although I guess you could generate some scripts in Photoshop that looks for common tones (like flesh tones) and equalize them. Not at all trivial though and very much a highly advanced thing to try. Your best bet is manual exposure compensation. Should be very quick. Just arrow through and correct either in quick develop with a few clicks or in Develop.
Try using the 'Auto' Tone button in the Develop module Basic panel. If the subject and background have near the same reflectance the exposure level should look uniform across the images. You can then use 'Auto Sync' to apply 'Auto' Tone to all of the image files and any other adjustments required. Images with subjects and/or backgrounds that have different reflectance may have to be manually corrected, but 'Auto' Tone may still be a good starting point.
I forgot to add that in LR5 if you hold down the SHIFT key and double click on 'Exposure' it will apply 'Auto' Tone for just 'Exposure.' Using SHIFT 'Exposure double-click with 'Auto Sync' set with will apply 'Auto' Exposure correction to each image individually, which is exactly what you are trying to do. See if that works.
For this kind of work you really need to take the flash unit off the camera, and use sync cable or wireless to a fixed flash unit. There's really no way around that.
Trying to fix it afterwards is always a ton of extra work and effort, not an efficient workflow at all.
Thank you all for your answers. I will definitely start by trying the Auto Tone feature, it sounds like that might help given the controlled conditions I am shooting in. As a side note, I believe the issue is related to the fact that my Nikon flash has an automatic protection feature that reduces flash output when the bulb is overheating. I'm shooting with off camera flash and the camera is on a tripod, so no moving parts. Just the flash putting out inconsistent light levels (which I guess is better than burning out the bulb).
I see, sorry for misunderstanding. If you plan to do this a lot you might want to consider one or two studio flash units, with a couple of umbrellas. A 250 watt unit isn't all that expensive (not much more than a higher-end Speedlight), and will give you absolutely consistent output, as well as a lot more flexibility.
As a side note, I believe the issue is related to the fact that my Nikon flash has an automatic protection feature that reduces flash output when the bulb is overheating.
Try setting the Flash to manual mode with reduced output selected and adjust the camera ISO and F stop for proper exposure. My Canon Speedlites provide manual settings in .3 EV increments all the way down to 1/64th output. A setting of -1 EV (1/2 output) may allow you to shoot continuously with no drop in flash output. You'll need to set the lens wider by 1 F stop or increase ISO setting 2x. As long as the subject remains at about the same distance from the flash (not the camera) there should be no need to change the flash or camera settings.
Match Total Exposures” is particularly useful if you use the exposure bracketing feature on your camera (i.e. the camera captures each scene multiple times at different exposures), as it allows you to match all the photos’ exposures with a few clicks rather than having to adjust each one by itself.
Here’s the low down: After selecting and adjusting the exposure of your base photo, use the Command or Ctrl key (depending on your OS) to select the other photos you’d like to match the exposure for. Then select Settings and Match Total Exposures to have Lightroom match the other exposures as best it can. Voila!
Could you post some screenshots.