Best Practice to Archive Old Data in MySQL
With InnoDB and PARTITIONs, you could set up daily PARTITION BY RANGE(TO_DAY(...)) and use "Transportable tablespaces" to disassociate a day from the table and move it separately.  This will be a lot faster than the queries it would take to do the INSERT...SELECT and DELETE.  Once detached, the partition (now a table in its own right) can be moved to another database or server without impacting the ingestion of more data.

One drawback:  InnoDB's disk footprint is a lot more than Archive's.

What kinds of queries do you apply to the logs?

Blogs:
partitioning
and
chunking lengthy deletes