  We explore a class of $\phi^{4n}$ 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, $\phi^{4n}$ with $n>1$, the
kink-antikink annihilation behaves differently from the archetypal $\phi^4$
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.
