First, there is no single item known as 'mulch' :-) Mulch is a generic term used to refer to anything that is applied as a protective covering to the soil surface. It could be bark, compost, your MIL's marble chips, shredded leaves, lava rock, old carpets, shredded tires, etc., etc., etc.

Second, any organic mulch never needs to be removed. And by 'organic', I mean a mulch that is made from something that was once living (bark, wood chips, compost, leaves, etc.) All you do is apply a new layer on top of the old - the old doesn't need to be worked in at all, although it won't hurt if it is. It will eventually decompose in place and add to the fertility of the soil underneath.

Finally, a non-organic mulch - like the marble chips, lava rock/gravel, shredded tires - can create some garden issues. Mulch made from shredded tires can leach heavy metals into the soil and damage plants. And rock or stone mulches can heat up in the sun and overheat plant roots and dry out the soil. They are also difficult to keep clear of debris and much harder to weed than an organic product. And finally, they do nothing to improve the soil - no additions of organic matter to stimulate and feed soil organisms, no nutrient contribution and no textural component to improve soil tilth.

Any mulch is better than no mulch at all but an organic mulch offers so much more than a non-organic one.

Very well said by Gardengal, above. Removing the organic mulch and replacing with any kind of stones would be a step in the wrong direction.