What is the best way to distance my relative from her forbidden relationship with a young man and prevent her from moving to his city, in order to protect her home and children from ruin and loss, while avoiding stirring up discord among siblings?

You have done well in advising this woman. Indeed, her relationship with this young man is forbidden and leads to corruption. The prohibition and sin are magnified by the fact that she is married, as this involves a dereliction of her duty towards Allah and towards her husband, whom she must protect in his absence, as Allah Almighty said: "So righteous women are devoutly obedient, guarding in [the husband's] absence what Allah would have them guard." Beware of having a bad suspicion of her, but if she persists in her error, you can threaten her with informing her husband. If it becomes necessary to inform him, let it be by hint so that he pays attention to his home. Islamic upbringing is the best means to protect homes from such evils.