Colors that exceed a smaller color space get lost with the conversion. There is not much you can do about it.
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If for example in Photoshop you use the Rendering Intent perceptual, all colors in the image get squeezed and shifted so that they fit into the smaller color space. This simple method keeps the overall impression of the different colors and often changes the overall contrast.
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With Rendering Intent relative colorimetric only those colors that exceed the target’s color space are affected. You compress and shift only this subset of colors, which are then moved to the edge of the destination color space. In other words, you only loose detail rendition for those affected color … and it is advisable to take measures if one wishes to preserve some of that detail → PL.
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In PhotoLab you can “desaturate” those affected colors, so that they fit into the smaller color space.
For RAW files and linear DNGs, we have the “Protect saturated colors” slider (Color TAB), otherwise in the Export window the “Preserve color details” checkbox – or the option to use “Soft proofing” with the ability to adjust the Preserve color details Intensity accordingly.
A much larger color space can create more problems than a less smaller color space. By example for me legacy=wide gamut + color rendering.
That is exactly what PhotoLab’s soft-proofing feature is designed for. – Before the conversion (Export), you get an on-screen simulation showing what the colors will look like after the conversion to the smaller color space.
Things are not made simpler or more reliable by discarding intense colors – such as by using DxO’s legacy working color space (ARGB) – but rather by reducing oversaturation in a controlled manner while preserving the necessary details.
This is for conversion to the sRGB color space (without a rendering intent in PL).
→ more on soft-proofing with a paper profile in my (next) reply to @zkarj 