We study numerically a two-dimensional random-bond Ising model where frustration can be tuned by varying the fraction a of antiferromagnetic coupling constants. At low temperatures the model exhibits a phase with ferromagnetic order for sufficiently small values of a, a<a_{f}. In an intermediate range, a_{f}<a<a_{a}, the system is paramagnetic, with spin-glass order expected right at zero temperature. For even larger values, a>a_{a}, an antiferromagnetic phase exists. After a deep quench from high temperatures, slow evolution is observed for any value of a. We show that different amounts of frustration, tuned by a, affect the dynamical properties in a highly nontrivial way. In particular, the kinetics is logarithmically slow in phases with ferromagnetic or antiferromagnetic order, whereas evolution is faster, i.e., algebraic, when spin-glass order is prevailing. An interpretation is given in terms of the different nature of phase space.