#JavaScript (ES6), 79 77 bytes
Takes input as (B)(A), where \$B\$ is a list of sets and \$A\$ is a list of lists. Returns a falsy value (false or undefined) if \$A\preceq B\$, or true otherwise.
B=>g=([a,...A],u)=>a&&B.every((b,i)=>u>>i&1|a.some(v=>!b.has(v))|g(A,u|1<<i))