A naive approach is to use the eigenvalue solution of your matrix $A(t)$ as the initial guess of an iterative eigensolver for matrix $A(t + \delta t)$. You might use QR if you need the full spectrum, or the power method otherwise. This isn't an entirely robust approach, however, as the eigenvalues of a matrix aren't necessarily close to a nearly neighboring matrix (1), especially if its poorly conditioned (2).
A subspace tracking method is apparently more useful (3). An excerpt from (4):
The iterative computation of an extreme (maximal or minimum) eigen
pair (eigenvalue and eigenvector) can date back to 1966 . In 1980,
Thompson proposed a LMS-type adaptive algorithm for estimating
eigenvector, which correspond to the smallest eigenvalue of sample
covariance matrix, and provided the adaptive tracking algorithm of the
angle/frequency combing with Pisarenko’s harmonic estimator .
Sarkar et al.  used the conjugate gradient algorithm to track the
variation of the extreme eigenvector which corresponds to the smallest
eigenvalue of the covariance matrix of the slowly changing signal and
proved its much faster convergence than Thompson’s LMS-type algorithm.
These methods were only used to track single extreme value and
eigenvector with limited application, but later they were extended for
the eigen-subspace tracking and updating methods. In 1990, Comon and
Golub  proposed the Lanczos method for tracking the extreme
singular value and singular vector, which is a common method designed
originally for determining some big and sparse symmetrical eigen
problem $Ax = kx$ .
: Comon, P., & Golub, G. H. (1990). Tracking a few extreme singular values and vectors in signal processing. In Processing of the IEEE (pp. 1327–1343).
: Thompson, P. A. (1980). An adaptive spectral analysis technique
for unbiased frequency
: Bradbury, W. W., & Fletcher, R. (1966).
New iterative methods for solutions of the eigenproblem. Numerical
Mathematics, 9(9), 259–266.
: Sarkar, T. K., Dianat, S. A., Chen,
H., & Brule, J. D. (1986). Adaptive spectral estimation by the
conjugate gradient method. IEEE Transactions on Acoustic, Speech, and
Signal Processing, 34(2), 272–284.
: Golub, G. H., & Van Load, C. F. (1989). Matrix computation (2nd ed.). Baltimore: The John Hopkins University Press.
I should also mention that solutions to symmetric matrices, such as what you must be solving given your use of
scipy.linalg.eigh, are somewhat cheap. If you are only interested in a few eigenvalues, you might find speed improvements in your method as well. The Arnoldi method is often used in such situations.