We provide new values for the model parameters of the covariant constituent quark model (with built-in infrared confinement) in the meson sector by a fit to the leptonic decay constants and a number of electromagnetic decays. We then evaluate, in a parameter-free way, the form factors of the B(B-s) --> P(V) transitions in the full kinematical region of momentum transfer. As an application of our results, we calculate the widths of the nonleptonic B-s decays into Ds-Ds+, D-s*D--(s)+ + Ds-Ds*(+), and D-s*D--(s)*(+). These modes give the largest contribution to Delta Gamma for the B-s - (B) over bar (s) system. We also treat the nonleptonic decay B-s --> J/psi phi. Although this mode is color-suppressed, this decay has important implications for the search of possible CP-violating new-physics effects in B-s - (B) over bar (s) mixing.