Aging LEDs - does their output drop after turn-on?
There are a lot of papers on LED aging (some on optocoupler aging, where the LED is by far the dominant factor), and it seems to be significant for IR to UV wavelengths, and accelerated by temperature. There are also papers on the optical "noise" from LEDs. The above-linked paper shows a very nice graph of 1/f noise down to 1Hz BW, extrapolation is risky though. 



I believe that variations in Vf with time would tend to imply that constant current will yield a more stable optical output. 

Last I looked in detail at this, many of the papers were looking at ways to reduce the aging effects so it's reasonable to conclude that they've probably been getting better, at least from reputable manufacturers. I believe there can be a large difference in LED aging between manufacturers, based on the optocoupler aging information I looked at some years ago, with the top-tier Japanese sources well ahead at the time. 

The venerable IL300 analog optical isolator uses two LED dice with feedback from a photodiode in order to stabilize the output of the two LEDs, with the implicit assumption that the light output of two similar LEDs operated at essentially the same forward current and under the same ambient temperature conditions will exhibit similar changes with regard to temperature changes and aging.