Let $D=\omega$ and $(x_0,...,x_n)R_n(y_0,...,y_n)$ just in case $x_0 <...< x_n$, $y_0 <...< y_n$, and $y_0 - x_n \geq n$. It is easy to see that for any two infinite sequences $s, s'$ there is some $n$ such that $(s(0),...,s(n))\not R_n(s'(0),...,s'(n))$ -- as $n$ increases $s'(0) - s(n)$ gets smaller. Thus $R^\omega$ is empty and $\Box \bot$ is valid in $(D^\omega, R^\omega)$. In addition, if $(x_0,...,x_{n+1})R_{n+1}(y_0,...,y_{n+1})$, then $(x_0,...,x_n)R_n(y_0,...,y_n)$ since if $y_0 - x_n> y_0 - x_{n+1} \geq n+1$.

Let $D=\omega$ and $(x_0,...,x_n)R_n(y_0,...,y_n)$ just in case $x_0 <...< x_n$, $y_0 <...< y_n$, and $x_n < y_0$. It is easy to see that for any two infinite sequences $s, s'$ there is some $n$ such that $(s(0),...,s(n))\not R_n(s'(0),...,s'(n))$ -- for some $n$, $s'(0) < s(n)$. Thus $R^\omega$ is empty and $\Box \bot$ is valid in $(D^\omega, R^\omega)$ though not in any $(D^n, R_n)$. In addition, if $(x_0,...,x_{n+1})R_{n+1}(y_0,...,y_{n+1})$, then $(x_0,...,x_n)R_n(y_0,...,y_n)$ since $x_n < x_{n+1} < y_0$.