How to leverage relationships from current internship for SA/FT recruiting / long-term career?

Currently doing an off-cycle internship at a reputable institutional LP (SWF/pension/endowment type). There isn't really a return-offer pipeline; they take multiple interns each year but only hire at the junior level every couple of years.

I've been fortunate to sit in on a number of meetings with my team and senior people (MDs/Heads of certain products/strategies) from BBs, MFs, and other reputable/global firms. Some of these are firms and teams I'd genuinely be interested in recruiting for down the line.

I'm trying to figure out how, if at all, I should leverage these interactions without crossing a professional line.

For example, one MD at a bank told me that I could reach out anytime during my internship if I had questions or wanted primers/materials on the space. We didn't have much of a personal conversation beyond introductions and shaking hands at the beginning/end of the meeting, so I'm not sure whether this was just standard relationship-management courtesy in front of my seniors ("we're happy to help the LP") or a genuine invitation to stay in touch.

A few questions I'm thinking about:

• Is it normal to follow up with the senior people I've met through these meetings? If so, would you use LinkedIn, work email, personal email, etc.? And would you reach out right after meeting them or wait until you actually have a reason to?
• Any thoughts on reaching out to other people on the team after meeting them through these meetings, or mentioning the connection if I end up recruiting with the team later? (Or do they talk to 100 LPs and it doesn't even matter?)
• Does any of this change if we ultimately don't invest with them, or if my own seniors find out I'm staying in touch with people I've met through these meetings? (I might be way overthinking this.)
• And generally, how would you turn these one-off interactions into actual long-term relationships? Not necessarily looking to ask for a job—just feels like a lost opportunity to meet these people and then never speak again.

I'm not necessarily looking to immediately ask these people for interviews/jobs either. If someone genuinely means "feel free to reach out," having an MD-level relationship where I can occasionally ask thoughtful questions and potentially develop a mentor/contact over time would itself be extremely valuable.

Part of me thinks the safest move is to do nothing beyond being professional in the meetings. But as an intern eventually looking for FT opportunities, it also feels stupid/huge lost opportunity to repeatedly meet senior people at firms I'm interested in, do nothing, and never build on any of those interactions.

For people who've been on either side of this—LPs or banks/funds covering institutional clients—what's considered normal? What would you encourage an intern to do, and what would make you think they crossed the line?

Would also love to hear any creative ways people have used these opportunities to build relationships over time.