As Kennedy tells the narrator, Amy's father was struck off his father's will after he ran off with the family cook.
She’s the daughter of one Isaac Foster, who from a small farmer has sunk into a shepherd; the beginning of his misfortunes dating from his runaway marriage with the cook of his widowed father—a well-to-do, apoplectic grazier, who passionately struck his name off his will, and had been heard to utter threats against his life.
The story becomes more relevant when Amy falls deeply in love with the mysterious Yanko. Nobody in her village agrees with the relationship, and most think that Yanko, a stranger from an unspecified country, is ruining the good nature of an unspoiled English country girl. When Yanko finally works up the courage to ask her father for his daughter's hand, Kennedy tells the narrator that Isaac Foster
contended that the fellow was very good with sheep, but was not fit for any girl to marry. For one thing, he used to go along the hedges muttering to himself like a dam’ fool; and then, these foreigners behave very queerly to women sometimes. And perhaps he would want to carry her off somewhere—or run off himself. It was not safe.
In light of his own elopement, Foster's reaction could be seen as highly hypocritical. However, the key phrase here is "it was not safe." He is worried that, like him, his daughter could become alienated from what she knows. Unlike him, however, she is already close to the bottom rung of society. Another step down could ruin her.