Mac Messages - deleting conversations

That has been discussed many times in this forum. You can only delete them one conversation at a time.

YOu can make a suggestion to Apple here:

Messages

Just the be sure I understand you intended Terminology " Conversation "

Can Right Click the upper Most Conversation and Delete the Entire Conversion from associated person

If one means a series of individual SMS message within the Conversation - previous posted covered that

refer below

One at a time is all I've ever found.

Re-reading and notice " It's taking a lot of time to clear out the inbox."

Now, that implies one is taking about E-mail in your e-mail client ?