I wanted to give you a quick option on this and the utility of the suggestion is dependent on several things. I have had to move data from SQL Server to MySQL often. I use a combination of BCP to get the data out and then the MySQL “equivalent” which is the LOAD DATA LOCAL INFILE.
It is easy to automate both pieces but that means you only after data and may not work if the number of tables/servers is large. It depends. Yeah, I got to say that!
Working with dates in MySQL can be challenging because SQL Server likes to export them in a default format that is not readily consumable by MySQL (in my experience). In short, you will have to play with the date formats if you do this to ensure you get both systems to play nice.
You can roll all that into a PowerShell script that is driven by an XML configuration file for the tables you want to work with and even do something similar with multiple servers. SSIS is the gold standard of ETL work but you have many options available.