This kind of deployment was originally developed to reinforce NATO troops in Europe, in the event of a Soviet invasion. Today, this kind of operation is considered normal. To pull it off, however, takes a lot of planning and practice.
The first phase takes place months or years before the aircraft fly off to some far off hot spot. This planning involves knowing what kind of airfields are available at the other end, as well as what sort of facilities (fueling, hangers, maintenance, access to military supplies, Etc.) For example, Saudi Arabia has a number of large and well equipped military airfields. The Saudis also use many American warplanes and keep large stocks of bombs and missiles U.S. aircraft can use. Thus U.S. air force units going to Saudi Arabia don't have to take as much equipment with them.
Flying to most places in South America, Central Asia or Africa are a different matter. Airfields and other facilities are meager. Thus the air force would have to move in a lot of tonnage in the form of fuel, bombs and portable shelters (for troops and aircraft.) But these areas are also tend to have a fewer hostile aircraft and fast moving armored units to threaten the air bases we are using. However, there would be a lot of light infantry armed with potable anti-aircraft missiles and other targets bombers have a hard time getting at. The air force would rather take on troops that drive armored vehicles and fly warplanes.
Once the decision is made to send warplanes overseas, several transports fly off immediately (within a day to two) containing the ADVON (advanced echelon.) The ADVON contains technical specialists, staff officers and liaison officers who make the final arrangements for the arrival of U.S. warplanes and their support troops. The staff officers make sure their guys can handle the targets selected and that all the supply, living and working arrangements are in order. The arriving aircraft need a place to park and space for maintenance. The pilots and ground crews need someplace to live. The ADVON also contains combat pilots so that when the American warplanes arrive, fresh pilots are available to immediately get into the fighters and bombers and fly off to do battle. The aircraft coming over will carry some missiles and bombs, so that when they land all they need to do is refuel (and perhaps add some more bombs) before joining the fight. Right behind the warplanes, air force transports bring over the maintenance crews and other support personnel.
The warplanes usually depart the United States a day or two after the ADVON. Flying off just before the warplanes are a lot of tankers (for in-flight refueling) and one or two transports carrying ESTs (Enroute Support Teams.) Actually, tankers will meet the warplanes enroute as often as necessary to keep everyone in the air. The ESTs carry maintenance specialists to deal with any emergencies (causing a warplane to land at an airfield along the way.)
Once the warplanes and their support teams are overseas, it's up to the logistics people to get enough bombs, spare parts and fuel arriving to keep the warplanes in action. Indeed, the main constraints on this sort of deployment is the availability of supplies and transport aircraft. The warplanes can move out very quickly, but the transports have to be brought in from elsewhere to load the support troops and their equipment. This type of deployment only works if there are enough supplies on the other end to keep the warplanes in action.
The availability of an airfield that can handle fighters and bombers is another risky proposition. While modern warplanes can operate from an airport that supports multi-engine commercial jets, the combat operations will restrict some of the commercial operations. The airport may even be shut down from time to time as a damaged warplane comes in for a landing. The U.S. air force has an excellent safety record when it comes to handling all those bombs and missiles, but the sight of all that ordnance might make local civilians nervous. The air force has engineers that can be flown in to a substandard air field and upgrade the place so it can handle combat operations. But this takes time, and transports as well.
But, as is often the case in warfare, getting there first with the most is often the margin of victory. Jet fighter-bombers are simply the modern version of sending the cavalry ahead to hold the enemy until heavier forces (ground troops) can arrive to finish the job.
One unique military development of the Cold War was the ability of combat aircraft to move from the United States to any part of the world and immediately (or within hours of arrival) go into combat. This means flying warplanes from North American bases to, say, the Persian Gulf and landing, getting a fresh pilot, some more bombs, fuel and then, within an hour or so of arriving, taking off to put the hurt on someone. During the 1980s, such movements were an annual exercise for most combat squadrons. For this reason, in the Summer of 1990, we were able to fly hundreds of warplanes to Saudi Arabia, and right into combat. OK, so they didn't go right into battle in that case. But all those aircraft were intimidating enough to keep Iraqi aircraft, and ground troops, out of Saudi Arabia. It worked.