The answer to your premise is "No." Crusader Kings 2 is a fun game, but it's still a game, not a history lesson.

Most european "mercenary" companies were simply veterans of existing conflicts who decided to attach themselves to a new campaign. In fact, many premodern professional soldiers fit the current description of "Mercenary," but there wasn't anything unusual about it at the time: all sorts of military companies and commanders opportunistically backed different sides based on plunder, risk/reward, and who seemed to be winning, which is why I don't really like the term "Mercenary" to begin with. In fact, prior to the Napoleonic Levé en Masse, most armies recruited whomever was willing to fight anyway. In fact, many early military adventurers were mounted men-at-arms; typically men who had been born into landowning families but perhaps stood to inherit little more than a horse and a name. These men were quick to attach themselves to powerful landed lords, offering their service in battle in exchange for lodgings and a place in their retinue.

So first recorded company for hire in Italy in the early 14th century was employed first by the Marquis of Monferrat in northwestern Italy (unsurprisingly, near France). The Monferrat was threatened on the west by the Counts of Savoy and to the east by the ruthless Visconti of Milan. Apart from calling his knights to arms to repel incursions by both the Milanese and the Savoyards, the Marquis Giovanni also hired what amounted to a band of Hundred Years War veterans laying siege to Avignon in southeast France. They were commanded by a German, Alberto Sterz, and an Englishman, John Hackwood. They numbered about two thousand men at arms, who rode to battle but "in the english fashion ("all'Inglese") dismounted to fight (probably due to fear of being unable to replace horses). They were probably also accompanied by archers armed with longbows.

Armed bands of men definetely could quickly become problematic when they were just milling about; in fact, banditry was a huge issue in France during the Hundred Years War. During lulls in fighting or truces, bands of armed men dispersed all over the country far from home and with no reason to exhibit restraint. In fact, pressing these men into regular armed service to give them something else to do was one of the driving factors behind the Estates General of France agreeing to fund a single professional Royal Army in 1439.

Individual bands weren't often large; Arnaud de Cervole was a particularly charismatic knight and petty lord in the French army and never personally commanded more than a hundred men. However, while pillaging after the Peace of Brétigny, Arnaud rounded up between two and three thousand soldiers and laid siege to Avignon!

Among these two or three thousand soldiers was the aforementioned John Hackwood. He and his companions had ransacked and occupied the castle of Jonville (possibly one of many places of that name on Burgundy) but quickly moved south. A few names of the knights who rode to Avignon with him are known: Bernardo della Sala, Robert Briquet, Esprit, John Creswell, Naudon de Bageran, Lamit, Bataillé, Esparre. Each knight had with him a contingent of men loyal to him, sometimes, as in this case, they sacked together, while other times they would work alone.

To raise the money needed to ransom Avignon, Pope Innocent appealed to several sovereigns. The King of France and Count of Savoy were uninterested, but the Marquis of Monferrat was, on the condition the bandits he paid off would fight for him. Some knights remained in France; Arnaud de Cervole was even hired by King John II of France to hunt down other brigands. John Hakwood and some two thousand men took up the offer, even devastating the town of Pont-Saint-Esprit when looking for other men-at-arms to join their company before crossing the Alps.

So to sum up, for the most part mercenary companies were unemployed veterans who moved swiftly from place to place, and suffered severe desertions if they could devise no ransom or find no employer (although they also faced desertions when they were employed; when several lowland Piemontese towns were occupied by Hackwood in the name of Monferrat and the Savoyards had withdrawn to the mountains resulting in a stalemate, several hundred english soldiers signed up for garrison duty in Crete on behalf of the Venetians). There are also accounts of companies "dispersing" after a particularly resounding defeat. Once the captain earned a new contract, recruitment could begin in earnest again.

In the politically fragmented Italian peninsula, between the 13th and 15th centuries, the Milanese, Paduans, Veronese and Florentines were almost continuously involved in some kind low-level warfare, often bullying smaller surrounding towns into submitting in order to establish a ring of fortified burghs around their capital. Oftentimes, these sieges were perpetuated by the mercenaries themselves, who needed to be occupied lest they start pillaging; perpetuating a vicious cycle by which larger, richer cities were constantly hounding out weaker ones.