Depending on the nature and amount of data, it might be sufficient to just have an API endpoint you can use to send input and update (and delete?) requests to, and then build a script on the "local" side to read your new data and call the API.
Or you could simply(?) dump your data to a CSV file, then post that to an endpoint that will either read/process that file, or just do a load data infile MySQL command. Alternatively, you could SFTP that file to the server, and have a cron job watching for new files and which would then process the CSV file when found.