We introduce the notion of families of $n$-marked smooth rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical varieties into the moduli space of $n$-marked abstract rational tropical curves $\mathcal{M}_{n}$.

We study the topology of tropical varieties that arise from a certain natural class of varieties. We use the theory of tropical degenerations to construct a natural, ``multiplicity-free'' parameterization of $\operatorname{Trop}(X)$ by a topological space $\Gamma_X$ and give a geometric interpretation of the cohomology of $\Gamma_X$ in terms of the action of a monodromy operator on the cohomology of $X$. This gives bounds on the Betti numbers of $\Gamma_X$ in terms of the Betti numbers of $X$ which constrain the topology of $\operatorname{Trop}(X)$. We also obtain a description of the top power of the monodromy operator acting on middle cohomology of $X$ in terms of the volume pairing on $\Gamma_X$.