In MySQL 5.6 the mysqlbinlog command allows:
– connecting to a remote server with  “-R”
– get all binlogs passing “–to-last-log”

To continuously dump the logs, use
– stay connected with “–stop-never” which implies “–to-last-log”

You can practice using this MySQL Community Docker Image, which provides all binaries and an easy master-slave setup.

You can try using this

# mkdir /tmp/test; cd /tmp/test
# wget https://raw.githubusercontent.com/ioggstream/mysql-community/master/ga/docker-compose.yml
# docker-compose -p test up # run master and server

# docker run --rm -ti --link=test_master_1:master --entrypoint=/bin/bash ioggstream/mysql-community -l

$mysqlbinlog -R  -h master -u root -p root  --stop-never master-bin.000001

Now just send some data to the master (eg. the sakila db)

For further info, check here http://dev.mysql.com/doc/refman/5.6/en/mysqlbinlog-backup.html