Merger timing while converting core system

This has been a big question lately, how can we handle mergers and bringing in new partners, while we are also planning on doing a complete conversion of our core system. We have ideas around holding off doing the integration until after the new system is in place, but we also don’t want to have a long time frame where we have the merging company still working on their old system and process and not being integrated into our process. Not sure if anyone has tried to do these large initiatives at the same time, but any insight would be beneficial.