When there's someone you know (first-degree connection) that knows someone you WANT to know (new contact)...

Talk to/pitch the first-degree connection - get them interested (or better, excited).
If they express enthusiasm, let them know you'd like to an intro to the person you're trying to reach and ask them if they'd be open to receiving a forwardable email request for an intro.
Email them a short thank you for the meeting with some key bullets recapping your conversation, asking for the intro to the person you’re trying to reach.
They should reply to you, copying in the person you’re trying to reach with a short intro for each of you.
Reply all, move the first-degree connection to BCC and start your email with “Thanks [first degree’s first name], moving you to bcc to save your inbox. Great to meet you [new contact's name]”. Proceed with your email asking for whatever consideration you need (keep it short).

In this way a) you have some credibly with the new contact bestowed by your contact b) your contact knows you’ve picked up the ball but is not inundated with a continuing conversation they don’t want to see c) your contact will get some social capital if the intro generates value.
The new connection will hopefully reply/continue the discussion without flooding your mutual connections inbox with the rest of the chain