Frequency and magnitude of power variations in a DC circuit.
In order to overcome these sort of sampling measurement errors you can make the problem "easier" by adding a large electrolytic capacitor on the power feed to the development board after the series measurement resistor. This is an "attempt" to measuring average current taken by the dev board. OK, I say you are measuring "average" but it depends how big your capacitor is in uF terms and how long your "violent spikes" are.

If you have a 1 ohm series resistor and a 100uF capacitor, the time constant of this is 0.1ms so you'll need more capacitance or more resistance. To get reasonably decent measurements go for a time constant of about 10ms - this should suit your multimeter.

As an aside, your multimeter may be perfectly capable of averaging over a 1ms period and giving you the results you think you may not be getting.