Why the question feels hard
I think it comes up because people see it as a salesy move. I was having a good talk, but now I need to pitch them, so how do I ask for a call to pitch them?
That's the problem. You shouldn't pitch them. If you're having a legitimate networking conversation where you aren't selling anything, just exchanging information, learning, and maybe helping each other, moving to Zoom is a natural next step.
Just ask
Say something like: Hey, this has been great. I really appreciate it. Would you have some time to jump on a networking call? It's that easy. Most of the time, if the messages have been going well, they'll be happy to.
Lately I'll sometimes add: no pitch, just wanted to chat. I didn't used to say that, but I'm more comfortable with it now. It makes them a little more confident they'll get what they're agreeing to.
Why the call matters
Getting off text and talking live changes how people respond. In a 2017 study in the Journal of Experimental Social Psychology, M. Mahdi Roghanizad and Vanessa Bohns found that requests made face to face were 34 times more likely to get a yes than the same request sent by email. The people asking didn't expect that difference at all.
A video call isn't the same as standing in front of someone, but it's a lot closer to it than a message thread. People can see who you are and hear that you mean it.
No bait and switch
Once you're on the call, continue the same conversation. The real trick to networking authentically is to be authentic and actually want to help people. At the end of the call, you can ask if they know anybody you should talk to for business. Just don't pitch them head on.
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