Household Wire Insulation (Asbestos Risk?)
I've been in the industry for over 40 years and have never heard of asbestos in the old wiring insulation. It is oil-impregnated cloth I believe. The more pressing problems that I see from your photo are that the cloth insulation may start to crumble off the copper conductors when you manipulate the wires and that the junction box appears to be overfilled. If the conductors have been run through metal conduit then you should be able to pull new wires if necessary. If the conductors are encased in armored cable conduit aka BX then you may need to call in a real sparky. If the box is indeed overfilled then you can install an extension ring and get creative with the fabrication of a little frame around the extension ring before installing the new fixture.