In all cases I carry a set of "Tyre Pliers", plug kit, core tool and spare cores, spare complete stems, tubes, patch kit and the compressor is supplemented by a manual air pump (which I pray to God I never have to use to inflate a tire!!!). Ratchet straps help immensely seating beads. Note I run steel "tube-type" steel rims so they don't have the dreaded "safety bead" feature that makes breaking a bead near impossible without a tire machine. Drive over these tires (with valve core removed) and the bead breaks just fine.
Safety beads on steel (or aluminum) rims are all that I will run (had too many tire failures at higher speeds; the safety beads help keep the tire remains from tangling about the axle, IMO). I also suspect that they help retain the tire when aired down, for additional traction, off road.
For people that do not want to carry (or do not have when they need it ) a Tyre Pliers bead breaker;
Field bead breaking is normally fairly easy simply: remove the valve core. get out the jack and position it between the tire(on its side) and the frame/bumper with the base of the jack touching the rim; then jack the bead off the rim.
Hi-lift jacks are good at this hydraulic jacks less so.
Once started step on the tire near the area where it has initially pushed past the safety bead, you can normally push it off by "walking" around the tire....
If anticipating field tire changing I carry a pair of tire irons ('way easier to use than jack handles, large screw drivers, separated spring leaves or pry bars) along with a drilling (smaller/more packable) or blacksmith's hammer (larger).
Normally, two 12 volt compressors are available to me; the portable for airing up tires and the backup (middle rated) ARB compressor that runs the air lockers (it is enough slower than the portable that it makes carrying the portable worth while.
Ratchet straps are a big help for initial bead sealing when re-inflating tires (along with any thing relatively viscous (dishwashing detergent, concentrated orange juice, etc.) to temporarily seal the bead, for re-inflation.
I haven't had to mount/dismount tires in the field since I gave up semi extreme wheeling with the "hey y'all watch this.." folks... the spare on the last Jeep went , I think, 12 to 15 years without touching the ground. (It was narrower & undersized to save weight (a "compact" 33x9.50 M/T spare).
Enjoy!