What would second and third sons look to inherit if anything, and if they weren't to inherit anything what would become of them?
Check out The Ties That Bound: Peasant Families in Medieval England by Barbara Hanawalt. It goes into considerable detail on this topic.
While primogeniture was the most common form of inheiritance, it wasn't universal. Many areas in England practiced what was known as "borough English," where the youngest son inherits.
Bear in mind, that this is simply the default. Peasants could and did write wills to account for alternate means of inheritance (often conditional upon the child providing for their parents in retirement). Even if a lord did restrict the division of a specific piece of land, there was also a fairly active market in land among the peasantry. Someone without surviving issue (or where their children had already been provided land through other means) could have exchanged land in return for a retirement annuity, or a family with only daughters could provide land as a dowry.
If there was absolutely no way to acquire land for a younger son, whether through inheritance or sale, the son became what is known as a cottager, effectively the peasant underclass. Wage labor was not unheard of for either farming or other trades, and many lords relied on labor paid daily rather than that required of their tenants.