Orchestral string players often have mutes that are clipped onto the strings behind the bridge, so they are quick to apply and remove, and don't get lost. One type has a small "handle" to make it easier to apply quickly. Another type just pushes against the face of the bridge, instead of being clipped on top of it (which is a bit more fiddly to do).
It's not an instantaneous operation, but 5 seconds should be enough time for a professional standard player. With the easiest type to operate
This video doesn't really show a "quick change, you can do" but it onegives a some idea of what is involved -handed while still holding the bow not much, and with the violin still under your chinafter a bit of practise.