While copying lists and libraries and selecting the option to preserve permissions, permissions will be assigned on the target library using the same permission levels as the ones used at the source. If you want to change the permission level that is used or if you have different permission levels in the source and on the destination, you can set permission levels mappings. Get started by clicking on Map permission levels.
In the "Permission level mappings" window you will see all source permission levels and their mappings to destination permission levels. By default, permission levels with the same name will automatically be mapped together.
In the right pane, you will find available permission levels on the destination site.
By dragging and dropping permission levels from the right pane to the left pane, you can re-assign permissions of lists and libraries that you will copy. Every place where the source permission level is used, it will get changed to the destination permission level during the migration. In the image below, every user or group with "Source Permission Level" access will have "Custom Permission Level" access on the migrated objects.