Long discussion, as I was away.
About up and down, it was about the levels of signals, going up and down much more than usually, say from quarter to quarter of an hour (each station behaving at its own pace).
About the Doppler effect, I am not sure, but the frequencies at 0.01 kHz seemed somewhat less stable, hard to tell from the usual drifting of many stations. But a 10 Hz (or more) Doppler shift would be quite unusual I think. But what can happen when there are ionized clouds/patches in the ionosphere ? Happily, a few stations seemed quite stable, so it's not a problem with my Rx (and I use the frequencies to 0.01 as help for identifying).