Move Project with Predecessors

I need help moving a project that has predecessors.  This is being used for a production facility to track the status of our projects.  However, I have run into a problem when needing to move a project in the order of production.  The predecessors are not moving with the project change. 

-I need the "Design" process of the next project to start one day after the "Design" process of the previous project ends. 

-I need the "Caseworks" and "Millworks" of each project to start once the "Design" process of its project ends.

-I need the "Finishing" of each project to start 1 day after the "Caseworks" begins.

-I need the "Hardware" of each project to start 1 day before the "Finishing" ends.

Is there anything I should be doing to make this happen?  Smartsheet1 is a screenshot of it before the move in production schedule.  Smartsheet2 is a screenshot of it after the move in production schedule.

Thanks for any help you can give. Betty