DMM: auto-ranging and capturing transient current waveforms
Well first of all, you'll need to give a specification for the DMM. Perhaps a high-quality lab multimeter could be useful, but in my opinion a simple hand-held DMM will almost certainly have too low sample rate to be able to capture what you want to see, even if you didn't have issues with range switching. Some multimeters might have sampling frequency as low as single digit value in hertz. If you take a look at the oscillogram, you can see that the scope is in 500 µs/division mode, so for example the pre-RX feature is around 100 µs long. I'm a bit skeptical of the possibility of capturing something like that with a multimeter.

One trick that's used on auto-ranging multimeters is to first manually check the ranges you have available and then switch to appropriate range by hand before measurement. Almost all auto-ranging multimeters will have option to select range by hand and this way, you won't have any issues with the gap that's created by range switching.
On the other hand, it could be that your multimeter doesn't have high-enough resolution in the high range, so you'll be missing out on the resolution with low current. High-count multimeters can help here, because their ranges will usually cover greater amount of values that can be displayed.