How to debug huge binary log file increases in MySQL replication setup?
I think your replication uses ROW as binlog_format. To verify, please run this on the master: 

SHOW VARIABLES LIKE 'binlog_format' 


The result would be one of the following values: statement, row, or mixed. More info.

In the case of ROW format, every changed row will add info to the bin log. If there are statements that affect a table of 1000000 rows, all these changes will be written to the binlog. 

For example, suppose you have a huge table, with 1M rows, and you run a statement like this:

UPDATE big_table SET hashed_email=md5(email)


This statement will add 1M entries to the binlog when it is row based, while it adds only one entry to the binlog when it is statement based.

Suggested solution:

Enable the general query log around the time when this this issue usually happens, and analyze all the statements to see if there are many that does update operation on huge number of rows.