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}]

Very nice, @Sean! There is simpler GraphEmbedding:

oldVertexCoors=GraphEmbedding[old];
newVertexCoors=GraphEmbedding[new];

And it probably worth fixing image size:

Manipulate[Graph[
  EdgeList@PetersenGraph[5, 2], 
  VertexCoordinates -> (oldVertexCoors + (newVertexCoors - 
        oldVertexCoors) t), ImageSize -> {300, 300}], {t, 0, 1}]

I'm not sure what it would mean to animate between graphs. Esp between ones that aren't isomorphic. You could, I suppose, take the position of each vertex in the graph and then interpolate between them in some way.

Do mean Graphs? Or do you mean plots, like "x-y" plots of some function?