Practicing exercises (specialy sweeping) should be done both clean to make sure you play each note clearly , and with lots of dist to take care of the muting stuff.

Develope your technique with 3-string sweeping first , cuz the muting job would be certainly be easier than more strings sweeping.

You should use both fretting hand and picking hand muting techniques , however , picking hand muting techniques are more usefull ... Take care when you use your palm to mute , that your not doing regular palm muting , you should play the note first , then kill it with your palm.