There is alot of misunderstanding of tyre inserts. Many think they primary function is rim defence well that part of what they do. In fact the charactetisics of the insert (size, responce to air pressure and the material itself) detemine how they work and what type of riding they are best suited too. 

So here is a detailed guide with some Physics. I used to be a physics teacher so I cant help my self. 

A tyre , insert and air system is essesially a spring with an overal spring constant.So the Insert really acts as a way of giving more control of the air pressure required to give the spring constant needed to stop your tyres botming out and squirming.  

Springs in series add in the following way. k(t) is the total spring constant of the system, k(a) is the spring contant of the air, k(r) spring contant of the rubber tyre and k(i) the spring contant of the insert. The spring contant of the air and tyre  are non linear, I.e non co stant with respect to displacement which complicates matters

1/k(t) = 1/k(a) + 1/k(r) + 1/k(i)

or k(t) = [k(r)k(i) + k(a)k(i) + k(a)k(r)]/[k(a)k(r)k(i)]

Now it should be clear the tyre used has a big influence on the pressure that can be run.

The spring contant of the air (low pressure) is the lowest term so 1/k(a) is relatively high compared to the tyre and insert term which are lower. So for tyres with flexible casing the air term does not dominate as much as trail tyres with stiffer casings. i.e you can run lower pressures in tyres with stiffer sdiewalls then with tyres with more flexible sidewalls. That does not mean however that inserts only work best with tyres with stiff sidewalls. Thats not the case at all.

The insert is important. The firmer the insert the more dominatant the air term becomes. If you have a 1kg trail tyre you will need a firmer insert to run the lowest pressures. In the PTN range the Rokkline work well but the Revolution insert is best. 

Run flat works best with stiffer tyres but the stiffer inserts (Rokkline and Revoluition) do allow run flat with light XC tyres. Most occasions where you will be running flat there will be some residual air pressure even if its 5 psi.

With a stiffer tyre and  an insert the air term dominates again in other words you control the feel of the bike with small changes in air pressure. 

The lower pressures can be be used allow lower rolling resistance. However the insert needs to reduce in size with air pressure and not be in contact with the tyre with the riders weight on it. Some insert are so big and stiff when the rider sits on them the tyre is in contact or the air volume so small (vittoria airliner) that there is very little tyre displacement before your tyre is in contact with the insert. That contact increases rolling resistance. 

If the tyre is not in contact with the insert then the insert is the last line of defense when you and or strike an obsticle preventing the tyre bottoming out on the rim and the consequential damage that can result. 

The lower tyre pressure improve grip and traction and as the tyre can conform to the gorund better you suffer lower vertical accelerations (bounced about less) and that is how rolling resistance is lowered. 

Essentially these insert work. Come insert like Crush Core or the vittoria and geared more to gravity riding as they are dense and very impact resistant. Crush Core is also heavy compared to PTN. PTN actually have a new heavier (slightly) and firmer insert called rockline which is aimed at enduro riders and E bikes. . Schwalbe pro core is quite good as you can alter the spring rate of the inner chamber to tune the system. However I have punctured pro core with a long thorn. The firmer inserts are better suited to trail tyres and the lighter inserts are better suited to lighter tyres. Many of the other inserts are heavy or heavy and dont offer good inpact resistance. The is a good article on this here.

So this is a suspension system in the tyre and should be viewed as such. Sorry it got all technical.

What inserts do a summary. 

• allow lower tyre pressures improving comfort
• reduce rolling resistance ( see above for explaniation and caveats)
• rim and tyre defense is a consequence of being able to run the lower pressure
• more traction and cornering grip for the same reason as you get lower rolling reistance. 
• Run flat potential.
• Anti burping