When you move a slider installation from one domain to another, you will need to update the skin's database information by clicking the 'Refresh All Skins' button inside Slider PRO -> Skin Editor.
Another important thing to know is that you shouldn't manually edit the slider's database content, such as image paths. The content inside the database is serialized, and trying to edit it directly in the database will cause some issues. Instead, when you try to update or replace image paths or any other content, I recommend using a specialized plugin, like Search Replace DB. This is a smart plugin that can replace data in the database even if that data is serialized or using another format.