I very nearly wrote my PhD thesis on this topic.
Here's as much as I was able to figure out, though it's hardly a direct answer to your question.
1) Say your total space is K\"ahler, and your fibers are compact. Then you can define a Levi-Civita connection on any open set consisting of smooth fibers. It turns out that this connection generates symplectomorphisms between the fibers.
2) In toric degenerations, the torus acts on the total space of the family, mostly moving them around, but preserving the zero fiber (which is why it's toric).
1+2?) Now imagine you use (1) to give a map from your general fiber $F_1$ to your special fiber $F_0$. Map further, to ${\mathfrak t}^*$, using the moment map on the toric variety.
Now you have an integrable system on $F_1$, stolen from $F_0$!
There's a problem: since $F_0$ isn't smooth, we can't actually use (1) to make the map. The hope is to take limits along the horizontal vector field to define a continuous function $F_1 \to F_0$.
3) It turns out that this is the same as following the gradient flow for the norm square of the moment map. And limits of real-analytic gradient flows on smooth varieties are well-understood, by Lojasiewicz. So if your total space is smooth, you can use this to show that the map $F_1 \to F_0$ is well-defined, continuous, and smooth away from the singularities in $F_0$.
I never got around to investigating how things change if the total space is singular (as in the Gel'fand-Cetlin-Sturmfels-Gonciulea-Lakshmibai degeneration motivating the questioner, and me too). Of course you can pick a resolution of singularities, and I guess you can
ask that the metric on the exceptional fibers be very very small, and use that to generalize Lojasiewicz' results. But I never worked on this seriously.
Example:
Let the family be $det : C^{2\times 2} \to C$. Then the $0$ fiber is the cone over $P^1 \times P^1$, so a toric variety, but the fiber over $1$ is $SL(2)$. That has a $T^2$ action, by left and right multiplication by its maximal torus, but doesn't have the rescaling action that the $0$ fiber enjoys. One can actually solve the ODE defined by the Levi-Civita/gradient flow and write down the map $SL(2) \to det^{-1}(0)$. It collapses $SU(2)$ to the singular point $0$.
What is the integrable system? Regard $SL(2)$ as $T^* S^3$, and the action variable as $(p,\vec v) \mapsto |\vec v|$. This generates unit-speed gradient flow on $T^* S^3$, which breaks down at zero vectors (the $SU(2) = S^3$) because they don't know which direction to go.
Watch out: toroidal $\neq$ toric !
It is not possible to realize this situation in a toric variety, at least not so that $\pi$ is a toric morphism, because toric varieties are rational by definition and complex tori are not. (I am assuming that by complex torus you mean a compact quotient of $\mathbb C^g$).
A toroidal embedding is an open subset $U\subseteq X$ in a normal variety $X$, such that for every closed point $x\in X$ there exist a toric variety $\overline T$, a point $t\in\overline T$, and an isomorphism of complete local $k$-algebras
$\widehat {\mathscr O}_{X,x}\simeq \widehat{\mathscr O}_{\overline T,t}$ such that the ideal of
$X\setminus U$ maps isomorphically to the ideal of $\overline{T}\setminus T$.
In other words, a toroidal embedding is something that locally analytically looks like the embedding of the open dense alberaic torus of a toric variety. I suppose the reference you are citing meant that it is possible to make the $\pi^{-1}(t\neq 0)\hookrightarrow X$ embedding to be toroidal.
I think that Theorem 2.1 of Weak semistable reduction in characteristic 0 by Abramovich-Karu produces a toroidal embedding for you. If not, then it should at least give you an idea of how to do it. In fact, section 1 of that paper collects the basics about toroidal embeddings, so you should check it out anyway.
Best Answer
The polytopes you are interested in are related by sequences of combinatorial mutations, as described here and here. If two polytopes $P_1$ and $P_2$ are related by a combinatorial mutation, then there is a construction due to Ilten (here) of a flat family $\pi\colon\mathcal X\to \mathbb P^1$ such that $\pi^{-1}(0)=X_{P_1}$ is the toric variety defined by the spanning fan of $P_1$ and $\pi^{-1}(\infty)=X_{P_2}$ is the toric variety defined by the spanning fan of $P_2$. You can interpret the toric degeneration of $\mathbb P^2$ to $\mathbb P (a^2,b^2,c^2)$ as a following a sequence of 1-parameter families by moving along the relevant edges of the Markov tree.