  We obtain exact solutions for kinks in $\phi^{8}$, $\phi^{10}$ and
$\phi^{12}$ field theories with degenerate minima, which can describe a
second-order phase transition followed by a first-order one, a succession of
two first-order phase transitions and a second-order phase transition followed
by two first-order phase transitions, respectively. Such phase transitions are
known to occur in ferroelastic and ferroelectric crystals and in meson physics.
In particular, we find that the higher-order field theories have kink solutions
with algebraically-decaying tails and also asymmetric cases with mixed
exponential-algebraic tail decay, unlike the lower-order $\phi^4$ and $\phi^6$
theories. Additionally, we construct distinct kinks with equal energies in all
three field theories considered, and we show the co-existence of up to three
distinct kinks (for a $\phi^{12}$ potential with six degenerate minima). We
also summarize phonon dispersion relations for these systems, showing that the
higher-order field theories have specific cases in which only nonlinear phonons
are allowed. For the $\phi^{10}$ field theory, which is a quasi-exactly
solvable (QES) model akin to $\phi^6$, we are also able to obtain three
analytical solutions for the classical free energy as well as the probability
distribution function in the thermodynamic limit.
