= Solution
Let $F:|K|\times[0,1]\to|L|$ be a <homotopy> from $f$ to $g$. The inverse images $F^{-1}(\operatorname{st}_L(w))$ form a finite open cover of the compact product. By the <Lebesgue number lemma> there is $\delta>0$ such that every subset of diameter less than $\delta$ lies in one of these inverse images.
Choose a <barycentric subdivision> $K^{(n)}$ with all its open stars of diameter less than $\delta/3$, and times $0=t_0<\cdots<t_m=1$ with gaps less than $\delta/6$. For each vertex $v$ and each time $t_i$, the set
$$
\operatorname{st}(v)\times J_i,\qquad J_i=[t_{i-1},t_{i+1}]\cap[0,1]
$$
(with the evident endpoint conventions) has diameter less than $\delta$ in the product metric. Choose a vertex $w_i(v)$ of $L$ whose open star contains its image under $F$, and set $h_i(v)=w_i(v)$.
At time $t_i$, the star condition makes $h_i$ a simplicial approximation to $F(\cdot,t_i)$. Adjacent intervals $J_i,J_{i+1}$ overlap, so choose a time in their overlap and a point in the interior of any simplex $\sigma$ of $K^{(n)}$. Its image belongs to the stars of every $h_i(v)$ and $h_{i+1}(v)$ for $v\in\sigma$. As in (i), their vertices lie in one carrier simplex, proving contiguity. Thus the sequence starts with an approximation to $f$, ends with one to $g$, and has contiguous adjacent pairs. Relabeling its indices gives the requested $h_1,\ldots,h_m$.
Here $\sim_c$ denotes contiguity, with the endpoint maps approximating $f,g$ respectively.
$$
\boxed{f\simeq g\ \Longrightarrow\ h_1\sim_c h_2\sim_c\cdots\sim_c h_m\text{ on some }K^{(n)}.}
$$
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