Best Way to Migrate Data

I have data like the following:
:

PK     Desc                          InsertedByUserID                                 UpdatedByUserID

1       Johnny                               13                                                        90

2       Frankie                               34                                                        75

3       Brenda                               75                                                       90

4      Scott                                   25                                                       14

The data is being moved into a new app.  The userid will be changed in the new app.  The initial migration of data is converting the old users to a new id. So I have no issues with the loading the new User table. The old id is being retained.  Every row of data in the database has an  InsertedByUserID  and a UpdatedByUserID column.  I have been trying to do the migration of data using a lookup of the old user id to the new user id. I am getting memory errors . I did also try a merge join of old to new on the insert id and another merge join for UpdatedBy id.  Seems a bit complicated. Is there a better way? I cannot use staging tables for the migration.  I cannot use linked servers either.  Sometimes there could be a few more ids in the data columns. 

So what is the best way? Please offer suggestions.......

DBA Developer