It seems a long time ago I tried to make some 'traditional' jazz bass mutes, gave up and so found a set of real ones, courtesy of a basschatter.
Recently the Squier jazz I fitted them to has been nearby, and it's reminded me that the sound is not what you'd expect. The sustain is about four seconds, not super long, but not the dull thump a chunk of foam under the strings or a clip on mute would give. The effect is partly losing the highs and partly to emphasisi the initial attack and first stages of decay. Great for a staccato sound (thin Andy Fraser on the Mr Big solo). It really changes how you play by making fast, complex passages really clearly articulated. It also brings the sound closer to a double bass than using flats alone would (I'm using them with roundwounds). I think I'll take it to my next rehearsal and see what happens.