A simple example:
Take a solar controller producing 100A charge, a 4000W Inverter/Charger and 1 100AH battery.
The SCC & Inverter are attached to 1 side of a Shunt, the other to the battery pack.
The SCC is producing 100A, the inverter is using 10A to power loads and so 90A is going to Batt.
Then you turn on a coffee maker that uses 50A, now the inverter will take 60A from the incoming power and the other 40A goes to the battery.
The coffee maker finishes, the inverter returns to 10A draw and 90A goes back to batteries. The Shunt registers the different amounts going to/from the battery. The incoming power source (SCC or Charger) is irrelevant as it just sees it as incoming power.
As a battery reaches full the voltage reaches a point at which the amps taken decrease's due to increasing resistance. The SCC is usually programed to change into Float Mode which trickles down till the threshold is reached and the battery is full. While the solar controller is in float mode, it can also increase the amount of power supplied IF there is a demand from the inverter. It can increase to the max capacity available (so it depends on time of day & production capacity). Again using the coffee maker example, if Batts are 100% full and you start the coffee maker & if Float can provide 50A extra it will, if it can only provide 30A then the balance 20A is pulled from the battery. After the maker is done, the float Amps can stay higher to replace what was taken from the batts and wind down amps as it completes.
Component Systems while slightly more complicated can be more efficient and further tailored to particular use cases. A good example of such is a part-time cottage/cabin where you want batteries maintained all the time but don't need a running inverter when not in use.
Below is a basic/typical component based system layout.
Hope it helps, Good Luck.