I would consolidate the Connection Managers down to a single set - i.e. remove the "Prod" versions. Then I would add SSIS Package Configuration to set the connection details, e.g. via an XML Configuration file for each Connection.
I would have a copy of the Configuration files for each environment e.g. in sub-folders. If they all need to be executed from a single machine I would swap the Configuration files as required.
Depending on your deployment scenario you might prefer the other Configuration options e.g. Environment variables, SQL - I think Config files are the easiest and most obvious to set up.