The slowly growing target mapping flags new source rows, and then inserts them to the target with a new primary key. The mapping uses an Update Strategy transformation to indicate new rows must be inserted. Therefore, when you create a session for the mapping, configure the session as follows:
In the session properties, click the General Options settings on the Properties tab. Set Treat Source Rows As to Data Driven.
In the session properties, click the Target Properties settings on the Mappings tab. To verify that the Integration Service loads rows to the target properly, select Insert for each relational target.