When importing fbx navigate to Animation section -> Extra Options and choose any Animation Take you want.
Import the fbx to C4D. Set the “Animate” Layout (upper right corner of the screen). Click on Takes tab in the animation window next to Timeline. You can see all the animations listed. Using the + icon next to every take you can switch between animations.
Import fbx to Unity. Click on the imported file. You can see all the animation clips in the Inspector-> Animation tab. Just click play to see how each looks.
Unreal Engine 4
When importing fbx make sure Import animations is ticked. You can see all the animations in the content browser.
Actions can be managed with the Action data-block menu in the Dope Sheet Action Editor header, or the Properties region of the NLA Editor.