I'm sure I must have gotten on this soapbox before?
Mentoring and networking are relationship-based and value-driven.
Seems obvious in theory but harder in practice. Relationships take time to establish trust and common goals and identifiers. Value is hard to define and very tied to an individual's goals. Usually, folks ask for mentors or start to network during a crisis when time is short and that's why the exercise gets the 'insincere' or 'grabby' labels.
I also read a lot about the process of mentoring and networking (sometimes I even write about it). But, like any process, when it hits a snag or a unique situation, it doesn't always leave you with any other tools at hand.
Basically, one falls back on relationships - the overlapping circles of your community that support you and are supported by you. The community you've been building for months/years. Everyone wants to help. Everyone wants to do the 'right thing'.... but when we're inundated with requests, we'll help those with whom we have a relationship of value first.
So practice your introductions. Know your goals. Make a long-term development plan. But understand none of this happens in isolation and all of it requires a community around you for success. So, first and foremost, build relationships - with genuine interest and caring all around.