Contradictions will be rampant as the vehicle and circumstance and requirements are so varied.
As slow as needed for the circumstances (Bumpy, rocky, eggs in transit etc.) and as fast as needed for enough smooth momentum.
This is aimed at heavy sand driving with a loaded overland vehicle.
A sand dune is a different story and special circumstance, same theory and logic, different requirement. You simply go for max in all of these. Max tyre deflation, speed, rpm.
This is an endless topic. One opens oneself to huge criticism because of unlimited opinions and experiences. There is also the endless combination of circumstances and vehicles. Therefore I will try to keep it technical and practical.
Smooth momentum is what you need. Momentum is generated in a couple of places. Speed combined with the vehicle weight is the one most people understand. Momentum also comes from engine torque combined with the gear ratio. This is affected by the torque at the rpm you are using. Big and older engines were designed to give massive torque at low rpm. The Cruiser 4.5 petrol normally idles at about 650 rpm. It will easily give you more than 310 nm at 500 rpm. The V6 torque curve starts at about 285 nm at 1200 rpm. The 4.2 diesel starts at 210 nm at 1500 rpm. The last 2 will more likely stall at a 1000 rpm should you suddenly floor the pedal. The newer versions may give even better torque but at substantially higher rpm. Diesels give torque at lower rpm but collapse when you go too low. Turbo diesels are worse at really low rpm, just see how easy they can be stalled when you pull away compared to a petrol engine.
You must have reserve torque available immediately should there suddenly appear a bad spot in the track. In a bad situation changing gears is normally not an option as you will immediately get stuck. (You have dropped the momentum) With the 4.5 you will have a good reserve driving at 1000 to 1200 rpm. With the V6 and the 4.2 diesel you will have to drive at about 1800 to 2000 rpm to get a similar reserve. The 4.5 can safely be worked in a higher gear and will most probably have a better fuel consumption than the other two because of that.
The main limitation here is that the engine must not labour at your chosen gear and speed. The best fuel efficiency normally comes with a load of 70 to 80 % of the torque at that speed. (Half throttle ?)
Gears. Gears as in high and low range and whatever else you have are just different ratios of increasing torque at the wheels. The gear should be selected to give enough torque and then a reasonable reserve for the bad spots. The worse the situation the bigger reserve must be catered for. (Higher speed or rpm and or lower gear) This is also influenced by the engine torque curve ! The higher the gear the better the fuel efficiency will be limited by, that it should not labour. Labouring also means you do not have a reserve. Test this by flooring the throttle. You must have a brisk reaction still and immediate speed increase without wheel spin. If not, gear down to increase rpm or up to reduce wheel spin. Lower gears have definite negatives apart from fuel consumption. (Consumption will change roughly with your rpm per km) Lower gears quite often have a high reserve of torque available and wheel slip is difficult to control with your foot. Higher gears are more gentle and controllable. They also allow a bigger variation in speed for your useable rev range. The rev range is rather limited with diesels so you must compensate for that as well. If you find a ‘sweet spot’ in 4th low range and another in 2nd high range, rather go for the high range. This will increase your reserves in momentum and speed and fuel.
Automatics should not be worked hard below 2000 rpm as the auto clutch will overheat the gearbox. Drive in gears with lock-up if possible.
In a 5 speed manual box 3 & 4 are best designed for continuous hard work. A little mechanical sympathy can go a long way. High gears and low rpm easily start a vibration in your drivetrain. This is highly undesirable. Immediately change to a lower gear. This is a loud noise, you cannot miss it.
Tyres and inflation. Tyres are the means by which the torque generated by the engine is transferred to the track to move the vehicle. Slip between the tyre and the track must be limited as much as possible. Different gears will change the amount of torque available. Too much torque will cause slip and dig the vehicle in. Deflating the tyres increases the contact surface and will decrease slip.
A tyre should be run at the right pressure for the load it is carrying. Bigger load, higher pressure. My rear tyres are normally at about 2,6 bar. Loaded for a three week overland trip it is normally at 3,5 bar. Lowering tyre pressure has a huge effect on your vehicle’s road manners. This is critical at higher speeds.
On a gravel road 80 km/h is a higher speed. To lower tyre pressure for the sake of comfort is downright dangerous. A little lower pressure than normal can sometimes increase grip an a corrugated road. Use with caution.
In sand decreasing pressure to less than a bar may sometimes be needed and the only way out. This is normally at pretty low speeds and road holding is not a crisis. This should not be done for long periods as heat will build up in the tyre and can destroy the tyre. Stop regularly and feel the tyre with your hand. When you cannot hold your hand on the tread for 30 seconds, the tyre is overheating. (Increase the pressure or stop for half an hour) Lowering the normally loaded tyre from 2 bar to 1 bar is the same as lowering heavily loaded tyre from 4 bar to 2 bar. Do not lower a heavily loaded tyre to 1 bar or less, you will destroy it. It is about the length of the footprint. A lightly loaded tyre will most probably have the same length at 1bar as a heavily loaded tyre at 2 bar. This can be measured on a hard surface. Test it for your particular loads and get a ratio before you go on your trip. Tyre diameter is the biggest positive factor and in sand the wider tyres have more negative effects than positive.
Tyre pressures in difficult situations should be controlled by, low enough to curb easy or constant wheel spin, otherwise as high as possible up to normal. That will leave you a reserve when you get stuck, to let it down to as low as 0.5 bar if needed. Pumped up, it will also keep your ground clearance as high as possible.
Going to Moremi for instance you can drive 95 % in high range. The 2 sand ridges and Savuti campsite you may need low range.
Some sand sense:
Never lock the brakes when you stop. Let it roll to a standstill in the last bit. Reverse a meter or more to give yourself an easy start. Stop on a downhill, it will be much easier to pull off. Keep a steady momentum going. Watch out for bad spots and get into the right gear before you get there. If you need to do a bad stretch, plan it for early morning, the sand is harder when it is cold. Let the trailer wheels down as well. Wide wheels look great, but they do not fit in most tracks as the locals and tour operators use standard width. If you tow get the trailer to the same track width as your vehicle. If you have different widths on the front and rear axle, add spacers to make them the same. Take a plate with for the jack, it will sink away in the sand. Get a decent pump to reinflate the tyres. Most cheapies will take as long as 20 minutes per tyre to get it to full pressure again. Even a hand pump is quicker. When you are out of the thick sand, reinflate the tyres before you start driving at speed. When you see a vehicle from the front and there is a spot to get out of the track, do so. There may not be one at the meeting point. Climb out of the track at speed. Do not stop and the try, you will get stuck. Only reduce the speed once you are out of the way completely. Practice this a bit as it can get tricky when the ruts are deep and the sand is loose.