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I'm not a Red user but a Sony EX3 for example also splits clips up clips. With this camera if you copy the whole folder structure Premiere at least sees clips which are split up but belong together as one. If you only copy the clips Premiere does not have the extra information for doing that. Beyond that you can mark all clips you want to treat as one and group them (by rigth-click > group) or you nest them them (right click > nest). Nesting is a more complex concept in Premiere which you may should watch a tutorial for if you are not familiar with it.
Nesting works better than grouping for cutting but note than rendering and encoding time might increase
Import spanned clips only via the Media Browser. All should be well then.