For official visits at D1 schools, typically the coach will bring it up at some point in the process, then nail down a date and eventually get the travel plans made. Pretty common to bring in a group of recruits on the same weekend.
The school usually pays for travel and the recruits stay with current athletes. There are ncaa limits on how long recruits can be on campus, I think 48 hours. The coaches will handle this.
So a typical program might bring in several groups of distance kids over several weekends in the fall, for example. Then there might be some recruits coming individually at various times throughout the fall and winter.
In terms of how coaches handle this and how many invites they extend, it varies a lot. Generally they’re narrowing the recruiting pool quite a bit, inviting a larger group than they think they’ll get, then making some decisions based on those visits. But they aren’t inviting hundreds of recruits; could be in the single digits at some programs to a few dozen at others.
There are some coaches who are pretty confident in their yield rate and keep visits pretty limited. Others bring in a lot either because they need to or because they like a lot of options.
Getting invited to an OV is a pretty big step. They use program resources and coaches don’t waste them. But it is not necessarily a guarantee of scholarship money or admissions support.
If this doesn’t come up in the course of discussions, I think it’s fine at some point to ask the coach if they bring people for OVs, how it works, when they decide who to invite. Just a simple question like, “might there be a chance to visit campus and the team when school is in session” is probably enough to get the ball rolling.
Some programs don’t have the budget or bandwidth to do many OVs so it’s possible some schools you look at might have a different process.