The color will always change when converting between color spaces unless all colors in the image are within the boundaries of the gamut of the target color space.
If you’re not doing it an ICC color-managed workflow, the transformations can be very jarring… clipping of out-of-gamut colors cause large areas of flat color where there once was detail and colors will shift, particularly RGB to CMYK.
Even when using an ICC color-managed workflow, you’re likely to get changes because while the spaces obviously overlap, clipping will occur for out-of-gamut colors, black generation will affect the more neutral colors and if you have GCR in your CMYK black generation settings it will affect all your colors.
I’ve been doing this a while and I haven’t found any fully automated method of “convert this image to this color space” that will render every single input satisfactorily. That’s why programs like Photoshop exist.