Ask a Automotive Brake Engineer

[Brakeman]

What, if anything specific, replaced asbestos in friction materials?
I change my own pads and am ambivalent about the dust.
Mesotheliomas are rare.
I really don't want one.

Asbestos use in brake pads is considered stone age. No company I know of uses it or has used it in the past 15 years or so. Even then it was just a carry over from the pre-thermal engineer days.
Asbestos is a good insulator. Unfortunately for the asbestos companies, we don't want thermal insulation, we want thermal conductivity (copper!). If the pads get too hot at the contact surface, they create a super hot plasma/fluid surface in which the friction value, mu, plummets. No one wants to have their brakes suddenly stop gripping at all because the surface of their pads got too hot. You want the rotor to take most of the heat away and the pads to take the rest of the heat away, but never do you want the pads to insulate the surface friction.

Unless you are going to be working on a 1930's truck restoration, you've nothing to worry about with modern brakes. Asbestos is long gone from the OEM market. Fly by night after-market pads? Well even them, probably nothing in the last ten years or so has asbestos.

Added:
Mineral wool is the current main replacement of thermal insulating material, it is used in moderation and does not have the health problems of asbestos.