Get rows inserted and updated dynamically in SSIS
It all depends on your SSIS version, but see Integration Services (SSIS) Logging in the official documentation.

Assuming a reasonably modern deployment I can show you an example of SQL 2012 Basic Logging for SSIS packages. It's very simple, as it is the default configuration, and all you need to do is drill down the provided reports by right clicking on your package and selecting the appropriate report.

First confirm your logging level in SSISDB:



Basic reporting will include a final rows written for your destination. It doesn't provide row counts throughout, if you want that level of detail either increase the logging level (will affect performance and storage!) or create your own solution.



You should be able to query all of the log tables and generate your own reports. Microsft just ships a pre-built set of reports that sit atop the basic logging.