Difference between preparing for Digital and Litho presses

I was in touch with a print company recently re. a job to be printed short run on a digital press, and, later, long run on a litho press. The company said to send in the same PDF files as their software would closely match the two presses.

On receiving the digital printed proofs and the cromalin printed proofs (processed through GMG colorserver software), the digital is 'bright', whereas the litho is 'rich' (darker, warmer). Both are very nice. The print shop said some adjustment can be done to the litho during print, but it would be hard to make the digital print look like the litho.

I've seen on the forum the question 'is it for digital or litho'? So I guess there's a difference in the preparation of files for both print mediums.

Assuming everything is done correctly (CMYK profile, soft-proofing, etc.) for the litho, what needs to be done for the digital to make the outcome as close as possible?

Thanks in advance,