In my company most Engineers travel not only to rescue AOG aircraft downroute, but also as flying spanners to bases with no engineering support and on high profile VIP flights etc. In addition, Engineers are based downroute to support various bases.
Generally, Line Engineers are used although Base Maintenance staff do sometimes get involved with the bigger AOG's
Volunteer's are almost always used so if for any reason you can't travel, there is usually someone who will!