If you use a 200mm lens on a crop frame you multiply it by approx 1.6 you get a focal length of 320mm, right? So if you go into the menu on a D600/610 and choose DX, then does that increase your focal length to 320 also even though it's a full frame camera? Also would you need to use a DX lens or can you use a FX lens and get the same effect? I'm curious.
Nikon DX sensors are typically near 1.5x crop factor, Canon might be 1.6x.
It does not matter which lens... the lens never changes in any way.... it projects the same image regardless of sensor size. So assuming using the same lens, the lens does not actually play any part in this. Only the scene view that the cropped sensor captures is different (the sensor is smaller, so its view is smaller).
A crop is just a crop. the image gets cut smaller, and so subsequently it has to be enlarged more later. This greater enlargement is a necessary part of the whole. It is the magnification that causes us to perceive a telephoto effect.
So, you can
A. use a DX camera sensor to crop the lens image (smaller than if an uncropped sensor didn't),
or B. you call tell a D600 camera menu to crop it smaller,
or C. you can crop it smaller any time later in a photo editor... any image any time later, cropped is always smaller.
All crops make it smaller, simply meaning you have to subsequently enlarge it more to see it larger later.... which is a magnification making it look bigger, comparable to as if a longer lens took an uncropped copy. You can see this in your photo editor by simply zooming in on the view of the image. Same thing then.
If assuming all other things are equal (i.e., 24 megapixel sensors, etc), then
The real difference is that
A. the smaller sensor crop with a 24 megapixel sensor will still have all the pixels, 24 megapixels.
The B. larger 24 megapixel D600 crop only has about 10 megapixels left.
the C. image crop later to same DX size would also only have about 10 megapixels left.
But the telephoto effect is true of any crop (crop smaller and then enlargement back to comparable size). It is just an apparent visual effect, but yes, your image does appear magnified (because the smaller image does have to be enlarged more). If you refused to enlarge it more, you would just have a smaller image (smaller cropped frame borders) still containing the same size subject object (not enlarged).
Of these crops, the way to bet would be A., still with 24 megapixels. Or a better way to bet would be an uncropped full FX sensor with a lens 1.5x longer, or standing at 2/3 the distance (same smaller views, but all the FX pixels).
This is NOT saying all these images are identical images. Distance changes Depth of Field, and where you stand changes perspective, and enlargement changes perceived sharpness (CoC). It is however saying the included scene borders are the same (same crop, same view of scene).