It depends how you're going to assemble it. If you're only using mechanical fasteners (screws/brads), you could safely dye or stain all the parts beforehand.
If you're going to use glue, you need to be careful about squeeze-out, as this can prevent the stain from penetrating the wood, unless dealt with correctly. This, in turn, will lead to uneven staining, blotches, etc. so mask relevant areas with tape.
Finally, how easy would it be to get at everything to stain when assembled? I'd guess that most projects are finished after assembly, although there's no reason why the components can't be done first - providing that you're careful not to get stain or dye on the mating surfaces of any glue joints. This is more important with spirit stains than water-based dyes, but even with water-based dye I wouldn't want any on the mating surfaces.
If you're going to stain or dye first, as well as keeping it off the mating surfaces, consider masking the internal corners before clamping, as this makes any glue squeeze-out easier to deal with. HTH
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