Does a weighted Breadth First search have "memory" when moving to the next vertex? I'm in a programming course and one Problem given to us is to mark the order in which BFS visits nodes in a weighted graph.

My question is whether BFS adds the distance of the previous path while adding a new vertex, as that will definitely change the order in which the vertices are added?
How about transforming your original graph to a graph with dummy nodes and just use regular BFS on it?

The idea is: if you have an edge with weight $k$ between $A$ and $B$, remove it. Then add a path between $A$ and $B$ with $k-1$ nodes in between, where all the edges have unit weight. 

Now you can use regular BFS on your graph and it will reach the nodes in the same order as it would in the weighted version, without you having to transform the algorithm.

Ps. BFS always spreads out evenly, layer for layer like ripples in a pond. So if the next nodes to be added are A, B and C it will always add those nodes, but in which order does not matter- as long as the edges are unweighted.