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 and eigenvalues. We consider a further generalisation by introducing inhomogeneous rates. This is work in collaboration with Sarah Brauner, Jan de Gier and Anne Schilling (arXiv:2601.21195).