[compuAI (OP)]

∇ΧE=-μ ∂H/∂t

Mathematically it is correct. With some remarks:
1. At the "pure field" level, where fields are not produced forcefully by motions of macroscopic objects, ∂H/∂t is the consequence, ∇ΧE is the cause, not vice versa, according to the principles of locality and causality.
2. H (or B) is not fundamental field, but vector potential A. We can also find some law how changes ∇·A with the time, as we did with curl (H or B).

[paul cotter]

∇•A=-εμ∂V/∂t

[compuAI (OP)]

∇•A=-εμ∂V/∂t

Exactly. At least in our Universe, as linear approximation.
In another hypothetical world can exist longitudinal waves mixed with transverse, instead of scalar potential:
H = 1/μ0 · rot A
d = 1/μ0 · div A
∂E/∂t = 1/ε₀ · rot H - 1/ε₀ · grad d
∂H/∂t = - 1/μ₀ · rot E
∂d/∂t = - 1/μ₀ · div E
energy density u = ε₀/2 · E² + μ₀/2 · H² + μ₀/2 · d²
enerdy flux S = [E X H] + (E · d)
∂u/∂t = - div S

Although I meant other thing:
∂/∂t (1/μ₀ · div A) = 1/μ₀ · div (∂A/∂t) = - 1/μ₀ · div E - 1/μ₀ · div grad a