A short update is appropriate, especially because the first consultant made the introduction. It does not need to become an ongoing reporting process.

Closing the loop shows that you valued the introduction and followed through. It also makes it easier for the first consultant to continue helping you because they know what happened after they connected you.

After the second conversation, a simple note such as, 'Thanks again for connecting us. We spoke today and your introduction was really helpful' is enough. You do not need to summarize the entire discussion.

Thank the first consultant after the introduction, complete the second conversation, and send one concise follow-up afterward. Continue updating them only when there is a meaningful development, such as applying or reaching an interview stage. If useful, this guide on consulting coffee chats gives more detail.

I would send a quick thank you/update. Doesn't need to become a whole email chain.