This can be done with manually updating the 3D positions that define the 1st mesh - (update the MeshGeometry3D.Positions).
But the problem is probably which positions to move. If you move all of them, then the whole mesh1 will be moved. You may want to move only the positions that face the mesh2. In this case do the following:
1) calculate the vector from mesh1 to mesh2 (vector = mesh2.center - mesh1.center).
2) for all three positions in each mesh1 triangles check if their normal vector is facing in the same direction as the previously calculated vector. In this case mark all three positions to be moved (do not move them immediately to prevent moving them multiple times when one position is used by multiple triangles).
This step requires that the Normals are set for your mesh1 - if not set them by calling Ab3d.Utilities.MeshUtils.CalculateNormals method.
The in a for loop go through the MeshGeometry3D.TriangleIndices and increase index by 3 on each for iteration. Read indexes for all three positions from 3 following TriangleIndices (var i1 = mesh.TriangleIndices[i]; var i2 = mesh.TriangleIndices[i + 1]; var i3 = mesh.TriangleIndices[i + 2];).
Now get the normals for all 3 indexes (var n1 = mesh.Normals[i1], ...), sum them and divide by 3.
This will get you the normal of the triangle (you could also calculate that from positons and use CrossProduct).
To check if normal is facing the same direction as the vector from (1) use Vector3D.Dot product with passing triangle normal and the vector from (1) to the function. The result is a double value - if it is positive, then the vectors are facing in the same direction; if negative then they are facing in opposite directions; if 0 then they are perpendicular.
3) Finally, move all marked positions.
I hope this will help you. If not, then please define your problem in more detail.
The new version of Ab3d.PowerToys that was released 2 days ago comes with a "Basic WPF 3D tutorial" that can help you get a basic understanding of the WPF 3D objects. You can also get some additional info from link on the Links page (https://www.ab4d.com/Links.aspx