I am interested in "heat-like" equations on $\mathbb{Z}$ defined by a transition kernel $K:\mathbb{Z}^2\to\mathbb{R}_+$, satisfying the following conditions for every $x\in\mathbb{Z}$: $$ \sum_y K(x,y) = 1, \qquad \sum_y y K(x,y) = x, \qquad \sum_y (y-x)^2 K(x,y) = 1. $$ In addition, I am assuming a support condition so that $K(x,y) = 0$ when $|x-y| > 10$.
The kernel $K$ defines an evolution operator (which I'll call $T_K$) on $L^1(\mathbb{R})$ via $$ (T_Kf)(y) = \sum_x K(x,y) f(x). $$
An example of such a kernel is given by $K(x,y)=\frac{1}{2}$ if $y=x\pm 1$ and $K(x,y)= 0$ otherwise.
I am curious about the properties of the operator $T_K^n$.
Question: Does the bound
$$ \|T_K^n f\|_{\ell^\infty} \leq C n^{-1/2} \|f\|_{\ell^1} $$
hold for the operator $T_K$ as described above?
I expect that one can choose the constant $C$ to depend only on the support condition assumed on $K$. Using the Fourier transform one can prove this bound when $K$ is translation-invariant, meaning it has the form $K(x,y) = \varphi(x-y)$. Aside from this case I do not know how to proceed (or if the bound I hope for is true!).