You have few option for automation:
You can use something like Laravel Envoy
Use Capistano (A tutorial on how to do it is here
You can create a Phing script to do that .This is an old example I did 2 years ago https://github.com/gafitescu/Phing-file-builds/blob/master/svn-export-ftp-copy-pack-file-and-db-dump/build.xml and https://github.com/gafitescu/Phing-file-builds/blob/master/svn-export-ftp-copy-pack-file-and-db-dump/build_dev.properties
Use Jenkins
Explained further here Deploying a Laravel website with Git
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