As I said, I kind of let the nature of the spirit involved, and how the mage addresses them to determine whether it needs to actually 'search' out a target or not. I also think the rule was purposefully vague for that reason as well. Let's the GM have a little leeway on what's required for any given search.
A water spirit *could* just be in connection with any pool of water in its range limit, so the hunt could be considered a success if the target is standing near something dealing with water (a fountain, a sink, a glass of water, etc.) so it wouldn't have to move... OR it could travel through the city's water pipes so if they're near a sink or toilet or something, they can be found, and the spirit had to actually follow a trail of some kind. This could also translate into why that particular spirit failed to find the target (must not be near water, or around a water pipe). Granted, it's more about the rolls and what is story appropriate- the rest is just flavor in how I gauge the mechanics, and interpret it to the player.
Leaving it up to the GM to determine the nature of *how* the spirit does it, also allows them to make a search easier or harder depending on the needs of the story (or the creativeness of the player in how they contact the spirit). The rest is just flavor in how I gauge the mechanics, and translate it to the player.
For Astral search, I agree with SSDR. The implication is that it is actively moving around the Astral looking for something. Being spirits, I don't consider them as (thematically) constrained as in physical reality, and they can just move around as they wish within range.
In either case, I personally calculate the range at the start based on where the mage had them begin the search. The bond between the summoner and the spirit will allow it to keep in contact with the mage regardless of where the mage goes after that, but once the favor is requested at X location, that area is where it focuses on.
Unless the mage wants them to stay in range of them while they search, but that's a whole different set of modifiers in my mind...
Crunch~