Breakers are made to trip within some time frame for a current value. For example, here is what a European standard "Type C" breaker behaves like :
Lets say the breaker is 10A
With somewhat exceeded current (typically overload or "weak short circuit" with bad contact and not very high current), it trips by the heating of a bimetal strip. The strip takes time to heat up, the higher the current, the faster it does. So the breaker won't trip immediately
For
10A breaker :
At up to 11.3A, it will never trip
At between 11.3 and 14.5A, it might or might not eventually trip, but anyway it will stay on for long time
Above 14.5A, it will definitely trip sometime..
At 15A it will trip in 1 min to 9 min
At 30A it will trip in 3 sec to 15 sec
At a little under 50A it will trip in 1 sec to 5 sec
At up to 100A, if the electromagnetic trip does not activate, it will trip in no more than 2 sec
With greatly exceeded current (typically "good contact" short circuit), the electromagnetic trip is activates and trips the breaker within about 1 AC cycle (though in marginally low currents, where the electromagnet only barely activates, it might buzz for a moment before tripping)
For type C breakers, this is between 5x..10x rated current. So, the 10A breaker will never "magnetically" trip at 50A or less, and will allways "magnetically" trip at 100A or more. What happens inbetween is up to the individual breaker, in some the electromagnet reacts to lower current threshold than in others. So, for a 80A one breaker might allready trip magnetically and another might still wait a couple seconds until the bimetal heats up
When breakers are added in series, there is some current in all of them
supply ------------/200A/------/15A1/------------/15A2/------------ light
If the only load present is the light, then its the same current in all breakers. If there are more loads connected, then there is some more current going in the 200A one and if something else plugged on the 1st 15A circuit, then some more current going through the 15A1 one too
If currents are within the "thermal" range of breakers, they heat up untill something trips. If for example the current is 50A through all breakers, then both 15A are heating up. One of them will trip sooner (differences between breakers, difference in ambient temperature in the rooms where the panels are). When it trips the current is stopped, so the other one stops heating and won't trip. For most purposes, it can be assumed that which breaker will trip is random. However, it probably will be allways (or mostly) the same one if you try again with the same breakers
If currents are within the "definitely magnetic" range of the 2 15A breakers (short circuit on a long branch circuit), electromagnets pull at the same time, and the mechanics of breakers start to move at the same time. I dont know about USA breakers, but with European ones usually both will be tripped, less commonly only one, and if one then it is random which one. But if the current is not too high, it is still in the "thermal" range for the 200A one so that one will stay on
If currents are within the "definitely magnetic" range of all 3 breakers (good connection short circuit between thick conductors), all 3 electromagnets pull and the mechanics start to move. It is possible to find all 3 breakers tripped
If currents are within the "maybe magnetic" range of the 2 15A breakers, it is possible that one will trip magnetically and one not
US breakers are more like European "Type D", where magnetic activation is in the 10x..20x range instead of 5x..10x. But otherwise the logic is the same..
A fuse have its blow time chart too, which more or less resembles the "thermal" part of the breaker's time chart, so the same logic applies. However, i think it is possible for a fuse to start to melt before the breaker trips and "saves it", in which case the fuse is going to blow faster or at lower current the next time
Arranging the system so the main breakers dont trip when the situation can be handled by the small ones is called selectivity. There are some considerations to it :
As a rule, it is more predictable what will happen in the "thermal" range of the breakers, as it takes time to heat. In the "magnetic" range everything happens too fast, and mechanics start to move at the same time before any contacts open
Estimating the max short circuit current available in a sub circuit. It is limited by resistance of the wires of the circuit (up to the point of short circuit). If the current is not very high (within 100's A), it is quite possible that it will be in the "thermal" range for the main. So you got what you wanted
Using breakers of different types. If you use Type D (10x..20x) for the main, it means that currents will have to be higher to trip the "magnetic" of the main. In the sub circuits you want faster breakers of Type C (5x..10x) or Type B (3x..5x). However, you should be carefull with the "high" types (D) since you extend the range of currents at which the breaker will not trip "magnetically" for a "not very strong" short circuit. Generally, Type D are not to be used in a home system in Europe
Using more advanced breakers, where there is better control over the timing in all current ranges. Those are bigger breakers, usually used in industry and for main in big industrial/commercial site panels