Average bounds for the $\ell$-torsion in class groups of cyclic extensions

For all positive integers $\ell$, we prove non-trivial bounds for the $\ell$-torsion in the class group of $K$, which hold for almost all number fields $K$ in certain families of cyclic extensions of arbitrarily large degree. In particular, such bounds hold for almost all cyclic degree-$p$-extensions of $F$, where $F$ is an arbitrary number field and $p$ is any prime for which $F$ and the $p$-th cyclotomic field are linearly disjoint. Along the way, we prove precise asymptotic counting results for the fields of bounded discriminant in our families with prescribed splitting behavior at finitely many primes.