We study how changing the weight datum $\mathcal{A}=(a_1,...,a_n)\in(\mathbb{Q}\cap(0,1])^n$ affects the topology of tropical moduli spaces $M_{g,\mathcal{A}}^{trop}$, $\overline{M}_{g,\mathcal{A}}^{trop}$ and $Δ_{g,\mathcal{A}}$ and the homology of the latter one. We show that for fixed $g$ and $n$, there are particular filtrations of these topological spaces which we can use to compute the reduced rational homology of $Δ_{g,\mathcal{A}}$ and the top weight cohomology of the moduli space $\mathcal{M}_{g,\mathcal{A}}$ of smooth $(g,\mathcal{A})$-stable algebraic curves.