I apologize for the delay in responding. No excuses, just the life of a DBA
The package I designed was for the destination server which means that it is assumed that the log backup and log shipping from the source server was already being managed through SQL Server, A third party backup software or through an SSIS package on the source server. The package I wrote takes files that were successfully copied and moves them to a new directory (on the same drive) for restoring. If you wanted to restore the logs within the same SSIS package, I would do the following:
Create a main SSIS Package to:
Execute the package (now a sub package) to move the log backup files (in this article) from the shipped logs directory to the Restore Directory.
Add a For Each Loop container that restores the transaction logs and then moves them into the Transaction log directory (for all completed restored logs).
Call this main package from a job. The job will not run unless the previous execution is completed.
If this is not what you were looking for please reply with new requirements.