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We explore a class of ϕ4n models with kink and antikink solutions that have long-range tails on both sides, specializing to the cases with n=2 and n=3. A recently developed method of an accelerating kink ansatz is used to estimate the force between the kink and the antikink. We use state-of-the-art numerical methods to initialize the system in a kink-antikink configuration with a finite initial velocity and to evolve the system according to the equations of motion. Among these methods, we propose a computationally efficient way to initialize the velocity field of the system. Interestingly, we discover that, for this class of models, ϕ4n with n>1, the kink-antikink annihilation behaves differently from the archetypal ϕ4 model or even the kinks with one long-range tail because there is neither long-lived bion formation nor resonance windows and the critical velocity is ultrarelativistic.