Are perlite and vermiculite sustainable?
For potting mixes, I have to use quantities of perlite and vermiculite. But I am concerned about their sustainability, and how they are obtained.
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Both perlite and vermiculite are open mined in various countries. Perlite is what's known as volcanic 'glass', whereas vermiculite is actually a mica like mineral. Perlite is also put through a heating process to persuade it to 'pop' in order for it to be useful in potting soil as well as other applications. Both materials have a number of other uses, notably in the construction industry; this link has a separate embedded link to information on vermiculite too: What Is Perlite?.
Ultimately, it depends precisely what you mean by sustainable; both products are mined all the time for other uses, but the heating process for perlite could be considered non environmentally friendly. In terms of horticultural use, there is a more environmentally damaging substance (when dug up) contained in many potting soils, which is peat, of which there is obviously also a finite supply. This For Peat's Sake: Conventional Potting Soil And The Environment gives a little more information on ingredients used in potting soils and their environmental sustainability.
perlite and vermiculite are both mined and then heated, so it's sustainability is suspect.
perlite industry says they offset carbon processing, which means they're not 100% green. they also say there is plenty and we will have it for generations to come, but that means it's finite and if you're looking into regenerative agriculture it is still taking resources away from the earth.
vermiculite's processing requires a lot of energy production from the heating and "exfoliating". also in the past vermiculite mines have had asbestos (why old insulated houses probably have asbestos) so even today it's recommended by the EPA to only use vermiculite outdoors or in ventilated spaces (1)(2). it also creates silica dust when manufactured, which requires active protection for the workers creating it.
I've found success replacing perlite+vermiculite with rice hulls, and peat moss with coconut coir. Sand also provides sharp drainage if that is what you are aiming for in your potting mixture. I amend the acidity of the soil with pine fine or other organic material and rely mostly on compost for nutrients. hope this helped!
Perlite is not a sustainable resource.
Perlite is a non-renewable resource. The world reserves of perlite are estimated at 700 million tonnes.
The confirmed resources of perlite existing in Armenia amount to 150 million m3, whereas the total amount of projected resources reaches up to 3 billion m3. Considering specific density of 1.1 ton/m3 confirmed reserves in Armenia amount to 136 million tons.
Other reported reserves are: Greece - 120 million tonnes, Turkey, USA and Hungary - about 49-57 million tonnes. 
Perlite world production, led by China, Turkey, Greece, USA, Armenia and Hungary, summed up to 4.6 million tonnes in 2018
To my limited knowledge, vermiculite is also not a sustainable resource. It is mined, and therefore is subject to overmining.
Although pure vermiculite has no asbestos, when it is mined (which is the only way people get it in volume), asbestos can be introduced:
Although not all vermiculite contains asbestos, some products were made with vermiculite that contained asbestos until the early 1990s. Vermiculite mines throughout the world are now regularly tested for it and are supposed to sell products that contain no asbestos. The former vermiculite mine in Libby, Montana, did have tremolite asbestos as well as winchite and richterite (both fibrous amphiboles)—in fact, it was formed underground through essentially the same geologic processes as the contaminants.
Pure vermiculite does not contain asbestos and is non-toxic. Impure vermiculite may contain, apart from asbestos, also minor diopside or remnants of the precursor minerals biotite or phlogopite.
Controversy over health risks
The largest and oldest vermiculite mine in the United States was started in the 1920s, at Libby, Montana, and the vermiculite was sold under the commercial name Zonolite. The Zonolite brand and the mine were acquired by the W. R. Grace and Company in 1963. Mining operations at the Libby site stopped in 1990 in response to asbestos contamination. While in operation, the Libby mine may have produced 80% of the world's supply of vermiculite.
The United States government estimates that vermiculite was used in more than 35 million homes, but does not recommend its removal. Nevertheless, homes or structures containing vermiculite or vermiculite insulation dating from before the mid-1990s—and especially those known to contain the "Zonolite" brand—may contain asbestos, and therefore may be a health concern.
An article published in The Salt Lake Tribune on December 3, 2006, reported that vermiculite and Zonolite had been found to contain asbestos, which had led to cancers such as those found in asbestos-related cases. The article stated that there had been a cover-up by W. R. Grace and Company and others regarding the health risks associated with vermiculite and that several sites in the Salt Lake Valley had been remediated by the EPA when they were shown to be contaminated with asbestos. W. R. Grace and Company has vigorously denied these charges.
The linked Wikipedia article, and its sources, have further details.