What do you call the act of searching for something by walking?
Combing (as a synonym for searching, not adjusting your hair with a utensil)
The police and dog spent the whole night combing the area
comb verb [T] (SEARCH)
› to search a place or an area very carefully in order to find something:
The police combed the whole area for evidence. Cambridge
Weird coincidence:
- Charlie Hebdo attack: Hundreds of elite armed police comb woodland ... Independent
- Armed police scour forest as Charlie Hebdo shooting... Telegraph
Also https://www.youtube.com/watch?v=MtkK3eijBso
COMB THE DESERT!
Nothing wrong with "searching" but if you want another word, I suggest "scouring".
- "The police and dogs spent the whole night scouring the area" sounds perfect.
scour (verb) "to look through (as a place) carefully or thoroughly in an effort to find or discover something."
- "The police scoured the city for the Typhoid Mary of this latest epidemic." MW
- " The detective scoured the scene of the crime for clues."
Police scour bushland for body after tip-off on missing boy. From ABC NEWS
Depending on what you're looking for, "canvass" could also work: the police canvassed the area. This specifically implies talking to people, however, in order to get feedback - so it doesn't work in a wilderness setting.
For walking, "patrolling" works best.
Patrol: the act of walking or going around or through an area, building, etc., in order to make sure that it is safe.
Hunt high and low (for someone or something) and look high and low (for someone or something);
- Fig. to look carefully in every possible place for someone or something. We looked high and low for the right teacher. The Smiths are searching high and low for the home of their dreams. (AHD)
The Free Dictionary