I can't clarify the actions of the sulfate/citrate components of the salts, but I do know that it is the magnesium that performs the relaxing and diarrhetic benefits.
I believe that the salt form affects how easily the body separates the magnesium from the carrier which would then affect bioavailability. Mg stearate is nearly inseparable and thus is often used as a supplement filler. Mg sulfate is easily separated and present in natural springs(important resources for sanitariums a century or more ago), thus accessible and considered safe, so that may be why it is most frequently used. And from my observation, since it was the form used 100+ years ago when these protocols were incorporated into 'treatments', naturopaths and the like of today, are reluctant to deviate too much from this path, but do acknowledge the usefulness of other forms of Mg.
I personally try to supplement with the malate form of magnesium during the week before a flush to get the benefit of the extra malic acid once the tablets are broken down.