Sorry for the late answer. I had to fix the issue the same day, leaving me no option but to create the missing classes by hand. This works but obviously isn't the way to go... I forgot about this question until I had the same issue... :)
I did an export of the module via Sites -> Export so that shouldn't be the problem.
The problem was about missing classes not the data, so moving data is irrelevant in this case. Custom modules can't be staged so that's no option as far as I see. The sealed checkbox shouldn't matter for this particular problem as well.
Thanks anyway both for helping.
After some more research I found out that you have to select the tables as well when creating your export. You have to select the correct module and the system tables related to this module. This was the reason my tables were not present on the new site where I imported the custom module.
So when creating your export, select the following:
- Global objects -> Development -> Modules -> [your_module]
- Global objects -> Development -> System tables -> [your_classes_related_to_module]