  Graph burning is a discrete-time process on graphs, where vertices are
sequentially burned, and burned vertices cause their neighbours to burn over
time. We consider extremal properties of this process in the new setting where
the underlying graph is also changing at each time-step. The main focus is on
the possible densities of burning vertices when the sequence of underlying
graphs are growing grids in the Cartesian plane, centred at the origin. If the
grids are of height and width $2cn+1$ at time $n$, then all values in $\left [
\frac{1}{2c^2} , 1 \right ]$ are possible densities for the burned set. For
faster growing grids, we show that there is a threshold behaviour: if the size
of the grids at time $n$ is $\omega(n^{3/2})$, then the density of burned
vertices is always $0$, while if the grid sizes are $\Theta(n^{3/2})$, then
positive densities are possible. Some extensions to lattices of arbitrary but
fixed dimension are also considered.
