$k$ary Shift Rule  Output  

Let $\Sigma_k = \{0,1,\ldots, k{}1\}$. Let $\Sigma_k(n)$ denote the set of $k$ary strings with length $n$. A function $f : \Sigma_k(n) \rightarrow \Sigma_k$ is a $k$ary DBsuccessor if there exists a DB sequence $\mathcal{D}$ such that each string $\alpha = a_1a_2\cdots a_n \in \Sigma_k(n)$ is followed by the symbol $f(\alpha)$ in $\mathcal{D}$.
Using the PCR as the underlying feedback function, the four binary DBsuccessors are generalized to a larger alphabet [GSWW20], each in two different ways. These rules test in $O(n)$ time whether or not a string is a necklace, which is a string that is lexicographically smallest amongst all its rotations.
The following generalizes the binary Granddaddy DBsuccessor, the lexicographically least $k$ary DB sequence. The rule is implicitly given in the proof of its corresponding concatenation construction [FM78], and also restated in [AAR+19b]. It can also be constructed via the Prefersmallest greedy approach.
If $\alpha = (k{}1)^n$, let $j=n$; otherwise let $j$ be the smallest index of $a_2a_3\cdots a_n$ such that $a_j \ne k{}1$. Let $x$ be the smallest symbol in $\Sigma_k$ such that $a_ja_{j+1}\cdots a_n x (k{}1)^{j2}$ is a necklace, or let $x=1$ if no such symbol exists.
$f(\alpha) = \left\{ \begin{array}{ll} x &\ \ \mbox{if $x \geq 0$ and $a_1 = k1$;}\\ a_1{+}1 &\ \ \mbox{if $x \geq 0$ and $a_1 \leq k2$;}\\ {a_1} \ &\ \ \mbox{otherwise.}\end{array} \right.$
The following corresponds to the $k$ary Grandmama DBsuccessor [DHS+18], which can also be constructed via a concatenation approach.
Let $j$ be the largest index of $a_2a_3\cdots a_n$ such that $a_j \ne 0$ or $j=1$ if no such index exists. Let $x$ be the largest symbol in $\Sigma_k$ such that $0^{nj} x a_2\cdots a_{j}$ is a necklace, or let $x=1$ if no such symbol exists.
$f(\alpha) = \left\{ \begin{array}{ll} 0 &\ \ \mbox{if $x \geq 0$ and $a_1 = x$;}\\ a_1{+}1 &\ \ \mbox{if $x \geq 0$ and $a_1 \leq x{}1$;} \ \ \ \ \ \ \ \ \\ {a_1} \ &\ \ \mbox{otherwise.}\end{array} \right.$
The following generalizes the binary PCR3 DBsuccessor. It first corresponds to the shift rule in [SWW17].
$f(\alpha) = \left\{ \begin{array}{ll} x{}1 &\ \ \mbox{if $x \geq 0$ and $a_1 = k{}1$;}\\ a_1{+}1 &\ \ \mbox{if $x \geq 0$ and $x{}1 \leq a_1 \leq k{}2$;}\\ {a_1} \ &\ \ \mbox{otherwise,}\end{array} \right.$
where $x$ is the smallest symbol in $\{1,2,\ldots, k{}1\}$ such that $a_2a_3\cdots a_{n} x$ is a necklace, or $x=1$ if no such symbol exists.
Perhaps the simplest $k$ary DB successor to state (along with PCR3 above) generalizes the binary PCR4 DBsuccessor.
$f(\alpha) = \left\{ \begin{array}{ll} 0 &\ \ \mbox{if $x \geq 0$ and $a_1 = x{+}1$;}\\ a_1{+}1 &\ \ \mbox{if $x \geq 0$ and $a_1 \leq x$;}\\ {a_1} \ &\ \ \mbox{otherwise,}\end{array} \right.$
where $x$ is the largest symbol in $\{0,1,\ldots, k{}2\}$ such that $xa_2a_3\cdots a_{n}$ is a necklace, or $x=1$ if no such symbol exists.
Other works that present $k$ary DBsuccessors include [ETZ86] and [YD93].
A more generic approach for any underlying nonsingular feedback function, is available for download (the underlying feedback function is hardcoded). The option available for generation above uses the feedback function $f(a_1a_2\cdots a_n) = a_1 + 1$. This implementation is based on [GSWW20]. It generalizes both the Grandmama and the PCR3.