Hi, I'[m currently learning Unnatural Selection by Muse, I'm at a fairly novice level of playing. A part of the main riff requires the palyer to go from 3rd fret/6th string to 3rd fret/4th string, and the same for on the 4th fret. It is in Drop D tuning so the 6th string vibrates more than in standard tuning. I'm playing this part of the riff with one finger to a fret and am picking the notes economy style. The problem I'm having is that I'm having trouble properly muting the string when I go from 6th to 4th and vice versa, and especially in drop D, the 6th string still vibrates when I go to the 4th string. The riff requires keeping a steady rhythm so I need to be able to mute that low D string inparticular without breaking rhythm and whatnot. How would I be able to get a clean muting technique that kills the vibration while still keeping the rhythm and then moving that finger down to the 4th string (and so on) the tab I'm using is this one http://www.ultimate-guitar.com/tabs/m/muse/unnatural_selection_tab.htm
Play it slowly, and gradually get faster. There's not much to say, other than you need to keep practicing.
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It would be easier to outside pick it in my opinion. Also, try moving the fretting hand closer to the neck, seemed to help me control the muting better when playing certain string skipped phrases.

Apart from that, everyone else is going to say play it slowly until you can do it, then up the tempo. Which is correct.
If I understand you correctly, you're saying that you're playing that part with one finger to a fret. You might want to try playing the note on the 6th string with your index finger and the notes on the 4th string with your middle and ring fingers. That way you can mute the 6th string with your index finger. Another option might be to mute the 6th string with your thumb (fret hand).