Measuring DC current in led constant current driver circuit

Quite possibly a bit of  both. Many LED driver designs are switch mode designs, where , the actual waveform has the correct average current but in fact oscillates with the switch  current being higher, and on and off with a mark to space ratio that makes the RMS current correct.  A typical DMM on a  1A to 2A full scale  range will drop about 100or 200mA  at full scale DC, but of course on a bursty waveform, the actual voltage drop during the active time of the current burst  will be higher. If the output voltage rises too high during the cycle, the driver will shut down,

You may get a better reading with an old fashioned moving coil instrument, that naturally does some mechanical averaging, or if you can force the DMM to use a higher range perhaps  say 10A full scale so the volt drop is lower.  Also make sure you are using the DMM correclty,  so it is not trying to read an AC or use the wrong input sockets for the measurement if it is that kind.

A quick sanity check of the meter's amps function may be a good idea, a loop in series  with a torch bulb and battery or other known more or less resistive load, if there is any doubt at all as to it functioning as expected,  I do occasionally  find mine has a blown fuse on the low amps range, usually after a moment of inattention during fault finding, or if my son has borrowed it.  ;-)

Mike.