HI Hokusai,
Thanks for the reply and the heads up about the Illustrator "effects" - that makes sense. I don't think I have used that many effects, if any, so hopefully I'll be OK in that department.
---
If bringing layers across from Illustrator is not an option it looks like my best bet is to make sure all my EPS files only contain the one page/project. I think I've only got this problem because I haven't got a layout programme like InDesign and I have used Illustrator for this purpose. I've never used artboards before but I'm hoping to get to grips with those in Affinity Designer.
I welcome any suggestions if anyone has any alternatives to saving each page individually - I have rather a lot of work in the EPS format so if I need to do anything to my existing files in Illustrator I'd need to do that before I move over completely to Affinity.