I work in printing and know both sides of the RGB - CYMK conversion process.
If you have not applied a specific printer's CYMK profile under color managment or soft proofing all the out of gamut warning is telling you is that the reds in the RGB file exceed the saturation of the most saturated red which can be produced with SWOP standard inks.
That's really not a big deal for one simple reason. The reddest red on the printed page will wind up being the reddest red overlapping Y+M pigments can print, and the eyes of the person viewing that photo, absent any other color reference, will accept that red object as being a highly saturated red. That in essence is what perceptual color is. Our eye adjust to reference tones and what our brains expect us to see.
So yes the colors will change when you convert from RGB to CYMK, but there's nothing which can be done in RGB to alter than outcome. The out of gamut colors in the RGB file will be converted to their nearest equivalents during the gamut conversion. You could also desaturate them in Photoshop, but that only changes the file to display the ultimate outcome of the color so you can predict how it will look, but it does not alter the outcome.
There are many technical factors incorporated into a good CYMK conversion which can only be determined by the company doing the actual printing by profiing each press / ink / paper combination. Color of the stock, reflectance, dot gain, ink trapping, under-color (UCR) and gray component removal (GCR) are all taken into account during the conversion. A printing company which has tight control over its color management will prefer to get RGB files and perform the conversion to CYMK themselves so all of those factors can be controlled an optimized.
Accurate, predictive proofing and color management evolved from the need for advertising clients, ad agencies, and printers to communicate color accurately. Cosmetic ad agencies where notorious for over inking proofs and using non standard inks to get deep saturated colors on the proofs they showed to their clients. The ad agencies would supply color separation films to the printers, who couldn't match the proofs with the standard web printing inks. Agencies and clients would compare the magazine to the proofs they'd supplied and demand a free re-run. That's why SWOP standard inks and proofing methods were developed.
Back in the 1970s when I started in printing management after a stint as a wedding shooter and National Geographic lab technician all supplied ads were accompanied by progressive (single color and combination) press proofs made with SWOP standard proofing inks. We'd measure the ink densities on the proof and inspect the halftone film and compare it to the dot structure on the printed proofs to verify that the supplied film would produce the same result on our press as the proof the client and agency had signed off on. If the evaluation determined we couldn't match the proof we would reject the supplied film.
Fast forward 35 years and now people compare an image on an RGB screen with the printed result. They will jump to the same erroneous conclusions about why the printer couldn't match the original unless they use soft proofing. What soft proofing is designed to do is to make the screen image look similar to what will come off the printer (darker and less saturated) so the client has a reasonable expectation of what those final results will be. Realistic expectations equal fewer complaints.
There are things that can be done to compensate for the inherent changes during printing. An increase in contrast can be used to compensate and fool the eye into thinking the colors are brighter, the same way increasing contrast on image borders fools the eye into thinking the image is sharper (i.e. USM). That's where soft proofing during editing is instructive, but its necessary to have a feed back loop. You need to see the results of the the on-screen edit in the final reproduction and then learn how to anticipate and compensate. It may be necessary to edit to create an image which looks over sharpened or too contrasty on screen to produce one which looks as good as an optimally edited screen shot.
If I were asked to supply CYMK files I'd run like heck the opposite direction because the person asking for them doesn't understand all the reproduction implications and will point the finger of blame at the files if their expectations are not met in the final reproduction.
CG