There isn't built-in functionality for this.
To do this, you would need to find the location of the vertices in both layouts and and interpolate between them.
Suppose you have two graphs:
old = Graph[EdgeList@PetersenGraph[5, 2],
GraphLayout -> "SpringEmbedding"]
new = Graph[EdgeList@PetersenGraph[5, 2],
GraphLayout -> "BalloonEmbedding"]
We can get the locations of the vertices:
oldVertexCoors =
VertexCoordinates /. AbsoluteOptions[old, VertexCoordinates];
newVertexCoors =
VertexCoordinates /. AbsoluteOptions[new, VertexCoordinates];
And then make a Manipulate that interpolates between these:
Manipulate[
Graph[EdgeList@PetersenGraph[5, 2],
VertexCoordinates -> (oldVertexCoors + (newVertexCoors -
oldVertexCoors) t)], {t, 0, 1}]