I've been playing a bit of Crusader Kings 2 and while accurate in some ways I've always found how mercenaries worked sort of odd. Basically you pay a bunch of money and an army far larger than anything you can levy suddenly appears at your capital.
Where would these troops stay? How did they store their equipment? How would local rulers react to what was essentially a small private army marching through their territory to another kingdom?
There are a lot of questions here, and it will vary by Company and period. However, the companies generally moved around wherever they could get paid. The late middle ages saw endless war between the Italian city-states, France, the HRE, and others. They often lived like an invading army, threatening cities, taking hostages and pillaging the country side. Accordingly local rulers were often hostile to these armies. Some companies were actually excommunicated by the pope and had a crusade called against them after they threatened Avignon.
I don't have time type out all the details here, but I highly recommend the book "Medieval Mercenaries" By Kenneth Fowler. It gives a chronological account of the life and exploits of the great Medieval Free Companies.