The slope, for drainage, of any walkway structure (and of the lawn,
wherever possible) should be away from the dwelling.

Sidewalks, to my recollection, are sloped 1/4" per 4' span (width).

The slope of the run should be away from any structure and the amount
of run slope may be dependent on the surrrounding lawn contours and
dependent on the amount of runoff guaged, calculated or anticipated
for the area, yet not compromise the comfortable/aesthetic use of the
walkway.

For a walkway not near a structure, a slight crown, in the center of
the walkway, is often formed, for drainage to each side.

*Our local city engineers claim "their standards" (sic), for public
projects, allow for 1' drop for every 12' run, when/where necessary
(sic). I've learned "when/where necessary" means (covering their ass)
when and where they screw up and have to manipulate the project
scenario to limit/reduce the screw up/damage to a minimum. Maybe the
professionals, who did your flagstone work when the house was built,
once worked for your local govt.

As Ed noted, I suspect some of the old work has likely settled/rose
and is now sloped toward the house.

Sonny