We will discuss de Rham and Hyodo-Kato flip-flopping for dual towers of rigid analytic spaces. The main tools are comparison theorems expressing the two cohomologies as pro-étale cohomologies of corresponding relative period sheaves that, by definition, satisfy pro-étale descent. As an application, we show that de Rham and Hyodo-Kato cohomologies of the finite level coverings of the Drinfeld space of any dimension d are admissible as representations of GL_{d+1}(K). This is a joint work with Gudio Bosco and Gabriel Dospinescu.