Online shortest path in ordered DAG
If I understand your problem correctly then you probably didn't find any research because the problem is easy to solve.
Unless I missed something, you want to maintain an edge-weighted &quot;layered&quot; DAG $G=(L_0, \cup \dots \cup L_k, E)$ (as defined in your question) and support the following two operations:

Init(): Create a graph with only one layer $L_0$, a single vertex $r \in L_0$, and no edges.

Query(): return $\min_{v \in L_k } d(r,v)$.

Add_Layer($L_{k+1}, E'$): you are given a new set $L_{k+1}$ of vertices and a set of weighted edges $E' \subseteq(L_k \times L_{k+1})$. The vertex set of $G$ is updated to $L_0 \cup \ldots \cup L_k \cup L_{k+1}$ and the edge set of $G$ is updated to $E \cup E'$.


You can do the above by storing: 1) for each vertex $v \in L_k$, the distance $D[v]$ form $s$ to $v$ in $G$, and 2) the minimum distance $d^*$ from $s$ to a vertex of the last level.
The init operation amounts to setting $D[r]=0$. To perform the query operation just return $d^*$.
To implement Add_Layer($L_{k+1}, E'$), you proceed as follows:

Set $d^*$ to $+\infty$,
For each vertex $v \in L_{k+1}$:

Set $D[v] = +\infty$
For each edge $(u,v)$ in $E'$:

Update $D[v]$ to $\min\{D[v], D[u] + w(u,v) \}$


Update $d^*$ to $\min\{d^*, D[v]\}$



Notice that this requires time proportional to $|L_{k+1}| + |E'|$, and is therefore asymptotically optimal. Notice also that this solution does not need to store $G$.