When you power up the LED emitter, do not read the voltage displayed on the supply. The current reading will be correct, but not the voltage because there will be significant voltage drop in the power leads (which will add to the total "forward voltage").
You can use heavier gauge wire and get a slightly more better reading, but it will still be less accurate. The best way is to use a separate voltage meter and touch the probe tips directly against the contacts on the emitter package, or very close by on the nearest exposed trace or pad on the star board (MCPCB). Just measure the voltage across the LED (in parallel and as close to the emitter as possible). The voltage measurement from your multimeter and the current reading from the supply will be much more accurate now. With this setup, just ignore the voltage reading from the meter.
If you do not have sharp enough probe tips, or you find it difficult to hold the probes, you can solder a wire onto the LED or star board. You can use a very thin gauge wire (thin magnet wire, a single wire from those old ribbon Parallel ATA cables, etc). The thin wires will put less mechanical and thermal stresses on the emitter, and they are easier to work with. Wire gauge will not affect the voltage reading since no current will be traveling throught the wire. The same goes to the power wires. You can use thinner gauge wire to transmit the power since your power supply will send the proper amounts of current no matter what (and your voltage reading is not as important any more).
With that aside, I just want to help save you the frustration if you are new to the PS and testing the LED in this manner (just in case). I was just wondering if you can adjust the current level (making the power supply both constant current and constant voltage). If so, be careful not to have the voltage on the supply too high (with no load) before attaching the emitter. Lets say you have it at 6V (by accident) before you connect the emitter. The power supply will adjust the voltage quickly, but obviously not quick enough for the LED, so the LED will see the 6V for a brief period of time. You can quickly toast the LED by sending too much voltage. I have done so when first testing my power supply (sort of intentionally on an old K2 just to see if it would send that short spike). Of course, 6V might not cause it to fail, and it may require over 10V, but it could (depending on the power supply). So set the voltage to the high end of the white LED range (~4V), and if it is not enough (due to the voltage drop in the thin power wires), then increase it to the point where your LED gets the proper amount of current. Finally, make sure the emitter is well heatsinked. Good luck with testing!
Cheers,
-Tony
'ed many a LED in the fashion you describe even with my adjustable current supply. What causes the failure though is not so much the time it takes for the supply to respond but the size of the supply output capacitor. The cap dumps its charge into the LED which probably does the damage. The cap also certainly slows the response of the regulator circuits of the supply
