Does After Effects handle color profiles properly? Final Cut uses ColorSync to convert between color spaces and color profiles. So it probably converts your source image to the color space you are currently working in (probably BT. 709), and when you export the ProRes file it will properly write the color tag to the file.
The thing is, a lot of video software (editor and players) just ignore the color tag and goes YOLO, guessing the color space or completely ignoring the color space info, before blaming Final Cut X (probably the first editor that finally color manages everything properly), you should check if After Effects and Premiere actually handle color space the right way or just show whatever they want.
This whole mess is the reason you have to still use an external broadcast monitor for color grading.