I sometimes hear a crow issue a call (pause) response (pause) call (pause) response before the caller goes down to the ground to forage. Whether they think of it in this way or not, do/can crows use the time between calls/responses to check to see whether there is a bird nearest to them that could alert them to a threat when they are on the ground? They would seem very vulnerable when on the ground, which is what got me thinking. I googled this, several times, but never was able to come up with anything.
Good question - I had a dive through the literature and found very few publications on the vocalizations of crow or other corvids. Apparently though, the features that you noticed have been identified by others and some meanings applied.
I think the calls you have noticed are called "counter-calling" (sometimes antiphonal calling) in the literature and seem to be largely territorial. As they are territorial, they do indicate in some manner, distance - if the calls were too close or the counter-caller visually identified, then there would be a territorial response.
Unfortunately, the best publication I could find on this is by Parr (1997)1 as a PhD thesis. I can access it through my institution, but others may not be able to.
She has this to say:
Crows separated by quite some distance exchange vocalizations in a series by series fashion, often matching both the type of series (usually medium caw or long- medium combination series) and the number of caws in it. Such countercawing usually is across territory boundaries.
She then goes on to say:
The final mode of vocalizing I will describe is taking turns in contrast. Here multiple birds, usually two adults, perch conspicuously at intermediate distance, approximately 25 to 50 meters, from each other on the same territory and vocalize loudly in turn with different kinds of caw series. Unlike duetting, each bird completes a series or two before the other vocalizes, and the birds are not in the same trees. Nothing obvious explains why the two use different caw series types. Neighboring pairs occasionally take turns in contrast at the same time so perhaps this is an unusually unhurried form of territorial duetting. Another possibility is that they are informing other family members of their whereabouts and different identities.
But this doesn't describe the behaviour that you are seeing, of a bird flying down to feed post-counter-calling.
In addition, a more recent publication by Mates et al (2015)2 found a similar reason for the counter-calling:
Calls with moderate PC 01 values were observed during “counter-cawing” between territories and are likely equivalent to Parr’s “medium caw” and Tarter’s “fading call,” calls of moderation duration and pitch that were also observed in that context. Parr found that crows responded strongly to and often approached playbacks of medium caws, providing additional evidence that they are associated with territorial advertisement.
Parr CS. Social behavior and long-distance vocal communication in Eastern American crows [doctoral dissertation] University of Michigan; 1997
Mates, E. A., Tarter, R. R., Ha, J. C., Clark, A. B., & McGowan, K. J. (2015). Acoustic profiling in a complexly social species, the American crow: caws encode information on caller sex, identity, and behavioural context. Bioacoustics, 24(1), 63–80. https://doi.org/10.1080/09524622.2014.933446