How can I map edgeweights from a matrix onto edges along the paths generated by another matrix

Update: "give different color to source (red) and sink (green) vertices"

ClearAll[map3]
map3[matrixA_?MatrixQ, matrixB_?MatrixQ, t1_Real, t2_Real] := 
  Module[{indices = Union @@ (Partition[#, 2, 1] & /@ 
     FindPath[AdjacencyGraph[Map[Boole[t1 <= # <= t2] &, matrixA, {-1}]], 
       #[[1]], #2[[1]], ∞, All])}, 
    HighlightGraph[Graph[DirectedEdge @@@ indices, 
      EdgeWeight -> Extract[matrixB, indices], 
      EdgeLabels -> "EdgeWeight", ##3], {#, #2}]] &;

Examples:

map3[matA, matB, .2, .3][{4}, {6}, VertexLabels -> "Name"]

map3[matA, matB, .2, .3][Style[4, Red], Style[6, Green], 
 VertexLabels -> "Name", VertexSize -> {4 | 6 -> Large}, 
 PerformanceGoal -> "Quality", ImagePadding -> 20]

Original answer:

Maybe something like:

ClearAll[map2]
map2[matrixA_?MatrixQ, matrixB_?MatrixQ, t1_Real, t2_Real] := Module[{indices = 
    Union @@ (Partition[#, 2, 1] & /@ FindPath[AdjacencyGraph[
       Map[Boole[t1 <= # <= t2] &, matrixA, {-1}]], #, #2, ∞, All])},
   HighlightGraph[Graph[DirectedEdge @@@ indices, 
       EdgeWeight -> Extract[matrixB, indices], EdgeLabels -> "EdgeWeight", ##3], 
     {#, #2}]] &;

Examples:

SeedRandom[4];
n = 10;
matA = RandomReal[{0.1, 0.5}, {n, n}];
matB = RandomReal[{0.1, 0.4}, {n, n}];

map2[matA, matB, .2, .3][4, 6, VertexLabels -> "Name"]

vLabels = {1 -> AGF, 2 -> CO12, 3 -> MA1, 4 -> MA2, 5 -> EGW, 
   6 -> CST, 7 -> WHS, 8 -> HOT, 9 -> TSC, 10 -> FIN};
map2[matA, matB, .2, .3][4, 6, 
  VertexLabels -> {v_ :> Placed[v /. vLabels, Center]}, VertexSize -> Large]

Row[map2[matA, matB, .2, .3][##, ImageSize -> Medium, 
    VertexLabels -> Placed["Name", Center], VertexSize -> Large] & @@@ 
 {{4, 6}, {1, 4}}, Spacer[10]]

Here is what I could do to answer my own question:

ClearAll[n, matA, matB, map];
<< IGraphM`;
<< BoolEval`;
SeedRandom[4];
n = 10;
matA = RandomReal[{0.1, 0.5}, {n, n}];
matB = RandomReal[{0.1, 0.5}, {n, n}];

map[matrixA_?MatrixQ, matrixB_?MatrixQ, lb_Real, ub_Real] := 
    Module[{matA = matrixA, matB = matrixB, t1 = lb, t2 = ub, 
    subgraphBetween, pindex, prule, paths},
    subgraphBetween[t1, t2] := 
    AdjacencyGraph[BoolEval[t1 <= matA < t2], VertexLabels -> "Name"];
    pindex = FindPath[subgraphBetween[t1, t2], 4, 6, Infinity, All];
    prule = DeleteDuplicates[
       Flatten@Table[  
         Rule @@@ Partition[pindex[[i]], 2, 1], {i, Length[pindex]}  ]];
    paths = Graph[prule, EdgeWeight -> Take[Flatten@matB, Length[prule]], 
       VertexLabels -> "Name", EdgeLabels -> "EdgeWeight"]
    ]

   map[matA, matB, 0.2, 0.3]

Although this partially answers my question, it is not so elegant. Partial because the respective elements from matB are not selected. I just selected an equal number of edge weights to have a working code. The correct selection of cells from matB should match with the edges in prule.