It doesn't really matter - completely an issue of personal preference :-) Spreading the new mulch over the old is faster, less work and will immediately revitalize the appearance of the garden, but you can mix together if you wish.

I always put the new right over the old.

on top

I put down chicken or horse manure, alfalfa pellets, rotted leaves, bone meal, seaweed, anything I have to improve soil and then add the new mulch. Nitrogen will help turn the old mulch into food for your plants. The earthworms turn the breaking down mulch into the soil. You don't have to do a thing except water now and then. Charles Darwin measured rocks sinking into the soil because of earthworm activity.

However, you shouldn't add fertilizers without a soil test to determine a deficiency. Nutrient overload is already a major problem in many urban and suburban soils.