Under US law, the question of whether a franchisor is legally responsible for damages caused by a franchisee generally depends on how much control the franchisor exerts over franchisees. If the franchisor exerts "enough" control, it is liable; if the franchisee is independent "enough", it is liable.
Courts use the law of principals and agents to determine who is liable. Because principals control agents, principals are generally liable for any damages caused by their agents. Thus, if the franchisor "controls" the franchisee, the franchisee is considered an agent of the franchisor, and the franchisor is liable for any damages she causes. If the franchisee is not an agent, the franchisor is not liable.
Whether someone is a principal or agent is often a matter of fact, of their actual relationship. That is especially true with franchises. Most franchise contracts explicitly say that the franchisee is not an agent of the franchisor. In these cases, courts look at the actual relationship to decide whether the franchisee is in fact an agent. The general rule is that a franchisee is an agent if the franchisor exercises "day-to-day" control over it. States differ as to what constitutes day-to-day control, so who is liable for what depends on which state it happened in.
Courts have also looked to see if the franchisor does things that create the appearance of control over franchisees. This appears to be the basis for the Vegan Whopper case. In its ads, Burger King said the Impossible Whopper was "100% Whopper, 0% Beef." For this to be true, Burger King had to have enough control over franchisees to make sure they kept the Impossible meat free. In other words, for the ad to be true, franchisees had to be agents, in which case, BK was liable.
For more details, see this article on Franchisor Liability for Franchisee Actions.
PS Thanks for asking an interesting question.