The Tsetlin library is a random-to-top shuffle process on permutations. A q-deformation using the Hecke algebra can also be implemented on flags over the finite field $\mathbb{F}_q^n$. The semigroup underlying the random-to-top Markov chain on flags is a left regular band, which provides a method to compute the stationary distribution. We consider a further generalisation by introducing inhomogeneous rates. This is work in collaboration with Arvind Ayyer, Sarah Brauner, and Anne Schilling.