I was recently editing a photo in ProPhoto working space on a wide gamut display calibrated to REC709 (I'm doing some video work).
I flattened the image and converted to sRGB space and noticed a color shift. The image is low saturation and should be completely in REC709 space. But there is a slight color shift upon conversion.
Why?
When converting from a wide gamut to a narrow one you may get different colors because of clipping. Conversion is done by a matrix and when the RGB values that occur after conversion are negative they are clipped to 0. This can cause slight shifts hue and significant decreases in saturation but also increases luminance and often that is the largest effect. The opposite can occur as well. When RGB levels are clipped at 255 luminance is decreased.
This effect is not observable when the monitor is in the same space as the conversion target since the clipping is the same for both. So, rather counterintuitively, folks that notice this are those that run wide gamut monitors. If one runs a profiled monitor at sRGB and converts to sRGB they will remain blissfully unaware of the changes.
What is printed from ProPhotoRGB is different. ProPhotoRGB is first converted to Lab and most, but not all, colors as ICCLAB limits a and b to values in the range of -128 to 127. This isn't the primary limitation as most realizable ProPhotoRGB colors can be defined in those limits. All , non-specular, reflected colors are within that space.
The problems with printing directly from ProPhotoRGB v first converting to sRGB (or any other matrix type space space) is that the path is different. After the color is converted to Lab, lookup tables map the color to what a printer can actually print and that can be a color that is outside or inside both sRGB and AdobeRGB. This process is completely different. Worse, the way colors are mapped when they are outside the printable gamut are undefined. Even for printable colors they are undefined using Perceptual Mode. It's basically up to the profile software peeps who try to create "pleasing" colors. At least it's within the context of a defined viewing environment for the more recent V4 profiles. It may, or may not match one's expectations.