The critical behavior of many physical systems involves two competing n 1- and n2-component order-parameters, S1 and S2, respectively, with n = n1 + n2. Varying an external control parameter g, one encounters ordering of S1 below a critical (second-order) line for g < 0 and of S2 below another critical line for g > 0. These two ordered phases are separated by a first-order line, which meets the above critical lines at a bicritical point, or by an intermediate (mixed) phase, bounded by two critical lines, which meet the above critical lines at a tetracritical point. For n = 1 + 2 = 3, the critical behavior around the (bi- or tetra-) multicritical point either belongs to the universality class of a non-rotationally invariant (cubic or biconical) fixed point, or it has a fluctuation driven first-order transition. These asymptotic behaviors arise only very close to the transitions. We present accurate renormalization-group flow trajectories yielding the effective crossover exponents near multicriticality.