IF you move the block reference using a transformation matrix, THEN it should move all the attribute references as well. TransformBy should be used for move and rotate all the time.
Example:
Dim point As Point3d = AbsoluteLocation
Dim matM As New Matrix3d
Dim pStart As Point3d = brLabel.Position
Dim vDiff As New Vector3d(point.X - pStart.X, point.Y - pStart.Y, point.Z - pStart.Z)
matM = Matrix3d.Displacement(vDiff)
brLabel.TransformBy(matM)
Also there are more advanced ways to do point/vector math built into the point and vector objects.
jvj