  We generalize the greedy and lazy $\beta$-transformations for a real base
$\beta$ to the setting of alternate bases
$\boldsymbol{\beta}=(\beta_0,\ldots,\beta_{p-1})$, which were recently
introduced by the first and second authors as a particular case of Cantor
bases. As in the real base case, these new transformations, denoted
$T_\boldsymbol{\beta}$ and $L_\boldsymbol{\beta}$ respectively, can be iterated
in order to generate the digits of the greedy and lazy
$\boldsymbol{\beta}$-expansions of real numbers. The aim of this paper is to
describe the dynamical behaviors of $T_\boldsymbol{\beta}$ and
$L_\boldsymbol{\beta}$. We first prove the existence of a unique absolutely
continuous (with respect to an extended Lebesgue measure, called the
$p$-Lebesgue measure) $T_\boldsymbol{\beta}$-invariant measure. We then show
that this unique measure is in fact equivalent to the $p$-Lebesgue measure and
that the corresponding dynamical system is ergodic and has entropy
$\frac{1}{p}\log(\beta_{p-1}\cdots \beta_0)$. We then express the density of
this measure and compute the frequencies of letters in the greedy
$\boldsymbol{\beta}$-expansions. We obtain the dynamical properties of
$L_\boldsymbol{\beta}$ by showing that the lazy dynamical system is isomorphic
to the greedy one. We also provide an isomorphism with a suitable extension of
the $\beta$-shift. Finally, we show that the $\boldsymbol{\beta}$-expansions
can be seen as $(\beta_{p-1}\cdots \beta_0)$-representations over general digit
sets and we compare both frameworks.
