Sounds similar to the German THW (Technisches Hilfswerk - engl. Federal Agency for technical relief) It's 99% (unpaid)¹ volunteers, and they assist in case heavy/specialized equipment is required. For example they build large pipelines and water reservoirs in case of wildfires to supply the firefighters, build tent cities including drinking water filtration, sanitary and kitchen facilities, as well as recovery operations after disasters².

While they mostly act locally (supporting buildings after fires, or recovering car wrecks when the firefighters don't have big enough equipment) they do have some international quick-response forces.

¹ The volunteers are insured via the state, and employers in non-critical jobs are required by law to release them in case of an alarm.

² They were initially founded as a civilian service to restore infrastructure and recover civilians during wartime, (Ziviler Bevölkerungsschutz - engl. civil protection) they thankfully haven't needed to do that job and nowadays focus on natural disasters, both national and intentional.