  In this paper we consider a time-continuous random walk in $\mathbb{Z}^d$ in
a dynamical random environment with symmetric jump rates to nearest neighbours.
We assume that these random conductances are stationary and ergodic and,
moreover, that they are bounded from below but unbounded from above with finite
first moment. We derive sharp on-diagonal estimates for the annealed first and
second discrete space derivative of the heat kernel which then yield local
limit theorems for the corresponding kernels. Assuming weak algebraic
off-diagonal estimates, we then extend these results to the annealed Green
function and its first and second derivative. Our proof which extends the
result of Delmotte and Deuschel (2005) to unbounded conductances with first
moment only, is an adaptation of the recent entropy method of Benjamini et. al.
(2015).
