Well, it is tough to get too specific without knowing what your source/destination servers are or the particulars of your package.
But you can define an expression on a connection manager, and you can use those to dynamically switch the source and/or destination server. If the server uses windows authentication, you can do an expression on just servername (assuming the same user has access to all servers). If not, you can do an expression on the connection string.
You assign a variable inside that expression, and then you either pass in that variable when executing that package (you would need separate jobs set up for each set of target servers), or you can create a database table on the ssis server with a list of source/destination servers (and maybe the time zone for those servers if needed).