The key thing to remember is that there was no single form of serfdom in Medieval and Early Modern Europe. Being a serf in 12th century France meant something quite different to being a serf (or villeins as they were called there) in 14th century England which in turn meant something very different to being a serf in 18th century Russia (Russian serfdom was by far the closest to the chattel slavery of the Caribbean, Brazil and the American South and by far the most brutal as Russian serfs could be personally bought and sold and their owners could punish them at will). As Stephen Rigby points out in his entry on serfdom in The Oxford Encyclopaedia of Economic History “it is such an elastic concept as to lead some historians to doubt whether the term has any analytical value … its precise form must be established empirically for each particular time and place.” I'll answer with reference mainly to the societies I'm most familiar with, Medieval England and France.

In France, being a serf from the 11th century on meant that your juridical status was that of an unfree peasant subject to the private jurisdiction of a lord. You worked on your land to produce for yourself (and no doubt for markets too, especially because lords preferred payments in cash than in kind) but the only courts you had recourse to were those of your lord, who held various privileges over you (i.e. monopolies on mills, wine presses etc) and could impose certain obligations on you like the corvee (unpaid construction work on roads, bridges, castles etc). The serf was thus bound to the lands over which their lord held jurisdiction, but if they ran away whether or not they would be returned would depend on whether the neighbouring lord chose to cooperate. For example, in a charter of 1144 King Louis VII of France returns a female serf called Havissa, who ran away to marry the mayor of Germignonville, back to the estates of the Benedictine Abbey of St Pere de Chartres. Other lords, however, might not co-operate and competitively offer lighter feudal regimes in order to attract them i.e. in a charter of 1105, the castellan of La Ferte, Hubert, attracts settlers to the new village he has created out of a cleared forest and the market town growing out of his castle by promising that they will be "free and unencumbered of all demands" except court fines and "if one of them will trade on my land, he shall render the just custom (some kind of market toll)."

In England, villeinage (the highly legalistic form of serfdom which developed with the creation of the national legal system, the common law, in the late 12th century) was less a jurisdictional thing than a tenurial thing. By holding certain lands from a lord you were obliged to perform a certain number of days labour services for him every year, your disputes had to be resolved through the manorial court and you had to make various payments to him. Villein status passed through the male line, through which the tenancy would be inherited, but sons who weren't inheriting (which sons depended on the inheritance custom of the manor) and daughters could leave, provided they paid fines to their lords (i.e. daughters paid the merchet to marry men from outside the manor). Alternatively a villein could simply run away to another manor, where he would be enrolled as a customary tenant rather than as a hereditary bondman, or to a town, where if he resided there for a year and a day he became free. Some lords would track down their villeins, but would then have to put them on trial in the manor court and respect the court's judgement i.e. John de Vere, Earl of Oxford, tracked down the fugitive Meryn brothers from his estate at Aldham and had them tried in the manor court to establish their servile status, with the jury (juries were always staffed by tenants) ruling in their favour. This was of course more than a century after the Black Death, which led to the decline and disappearance of serfdom in England because, due to a strong market economy having developed in England by 1300, when the population drastically fell the value of land went down and labour went up, so lords had to compete for tenants on favourable terms otherwise the system risked collapsing. I'd highly recommend reading "The Decline of Serfdom in Late Medieval England" by Mark Bailey as its a very good up-to-date take on the subject that synthesises all the best research on the rural economy of late medieval England done over the last 40 years.