What do you need your transformation for? One possible transformation would be $f(t)=A^{1/2}(A^{-1/2}BA^{-1/2})^tA^{1/2}.$ It satisfies $f(0)=A, f(1)=B$ and is the geodesic w.r.t. to the metric $d(A,B)=\|\log(A^{-1/2}BA^{-1/2})\|_{tr}$ where $tr$ denotes the trace norm, i.e. $\|A\|_{tr}=trace(AA')$, see also "The Riemannian Geometry of the Space of Positive-Definite Matrices and Its Application to the Regularization".

Regarding the transformations (1) and (2): First note that the operations permute. Second note that if all eigenvalues have multiplicity one then the function which assigns the eigenvalues and the functions which assigns the eigenvectors are continuous. Hence in this case the conjecture holds true. If a function has multiplicity more than 1 it gets more complicated...