I seek to find any research that exists on the relationship between 
how heavy a man is and how deep his feet go down into dirt, sand, gravel,
loam,clay,etc.

Would this not also depend on the size of his feet?

It would also depend on his gait.  If he walked flat-footed  - not
coming down with his heel first -  he would not sink as deep.  If he
were running, his intertia could increase the depth of his footprint. 
It is also possible to lessen the depth for a couple of steps when
moving quickly by letting inertia carry one forward without putting
one's entire weight on those steps.

Too many variables, at least at this stage.

Good points, Myo.

I would also guess that, weight-for-weight, a foot stomping Irish
dancer would create deeper footprints than say Fred Astaire who, as we
all know, was so light-footed that he was even able to dance on the
ceiling, as he demonstrated in 'Royal Wedding'.

This topic was featured on a recent episode of CBS's Numbers where
they used a mathematical formula to estimate a person's height and
weight according to footprint depth. I am unsure how much of this was
fiction. I would suggest contacting a crime lab if you are really
interested as there doesn't seem to be any info on the web regarding
this topic.

Have a good day.

I suggest that because of the many and varied types of surfaces where
footprints will be found that you are very unlikely to find a
definitive work that is able to say that "If the footprint is this
deep then the person is that heavy" for each of the types of surface.
There are too many factors. Take sand alone. What type of sand? How
compact was the sand? Was the sand dry or damp or wet or saturated?
The comments made by probonopublico, myoarin & thither also identify
relevant factors.

It is my supposition that forensic scientists are most likely to test
the soil at the individual scenes of crime on a case by case basis and
perhaps apply known weights, etc. to the same soil, measure the
results and compare to the footprint in question using, as necessary,
the instruments of engineering & science such as soil compaction
meters, hydrography meters, etc.

I put the two words "footprint" and "forensic" together into google
and found many articles and references. Skimming the references shows
it is clearly a well developed area of forensic science. Most interest
appears to be in both shape of footprint and the patterns of footwear,
socks, stockings etc. and Also in the relationship (correlation)
between the length, breath, surface area of a foot (and therefore a
footprint) to the height and weight a person as these are not
dependent on the surface where the footprint is made.

There is also clearly links between research in the medical science of
podiatry and the forensics of footprints.

From google I merely select out the following samples:-
http://forensic.to/shoeprint.html
A page on the web-site of Zeno Geradts a Dutch forensic scientist.

http://www.latrobe.edu.au/podiatry/forensicpod.html
This appears to be a course outline for a "Forensic Podiatry" unit at
Latrobe University in Australia that includes a brief list of
recommended reading
One of the books stood out to me merely because of it's title
"Robbins, L. M. 1985. Footprints: Collection, Analysis, and
Interpretation. Illinois: Charles C. Thomas."

http://www.joannabriggs.edu.au/pacesetters/Vol2_Iss1/2_1c_sjones.html
Brief biography with quotes aboout Dr Sara Jones "...a podiatrist with
a passion for forensic podiatry..."

http://www.usc.edu/CSSF/History/2002/Projects/J1003.pdf
A abstract titled "Tell Tale Footprints: Determination of Stature from
Foot Length in Forensic Cases"

Hope this helps :)

Wow, ponder852, you have been doing a lot of pondering.

Great Comment!

I see that you are a recent inhabitant of this here GA-laxy ...

Welcome Aboard!