A reminder unemployment and underemployment and labour displacement existed in Roman times, and could be inferred to have carried into post Roman serfdom and the age of kings. It might not be the best choice for a peasant normally but walking off the land did happen. There are court records seeking the return of successful townspeople provably off their lords domain, and similar documents around marriage and land inheritance.

Peasant revolts would be fights for retained rights, even if informal - not just new rights, if at all about new rights.

Labour mobility predates the modern era.

You’re right that "walking off the land did happen". Although the feudal system legally bound serfs to the land, there were ways for individuals to escape this bondage. For instance, a serf who lived in a town for a year and a day without being reclaimed by their lord could often gain their freedom. These people, known as 'villeins' in some records, were essentially free peasants who had successfully left their lord's domain.

A boon for the very thing you describe was the Black Death of the 14th century. The colossal depopulation of Europe increased the de facto bargaining power of peasants to the point where they could competitively seek out better labor wages and land concessions in places well outside their previous manor. Thus, many peasants did exactly this, to the point where the noble elites and monarchies of Europe tried to enforce tougher regulations and laws against free movement, wage increases and even conspicuous displays of prosperity by the increasingly wealthy peasant classes of society (many of who were also turning to mercantile ventures to further diversify their income.

As is usually the case with government social and economic dictates that attempt controls against the practical social and economic reality of the world around them, these laws slowly but inexorable failed, leading to the steady erosion of feudalism throughout Europe (though not everywhere at similar times, and in some places this repressive system lingered for centuries longer, ie: Russia, Sicily, etc)

If anything, for all its grim deadliness, the Black Death was oddly beneficial to the future social and economic flourishing of Europe, starting with the rise of the Renaissance, and leading from there to so many other things, for better or worse for the rest of the world.