$k$ary Successor 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 $\mathcal{D}$.
DBsuccessors based on the PCR
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 successor 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].
Generic constructions for arbitrary nonsingular feedback functions
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 $f(a_1a_2\cdots a_n) = a_1 + 1 \bmod k$. This implementation is based on [GSWW20]. It generalizes both the Grandmama and the PCR3.
 [AAR+19b] G. Amram, Y. Ashlagi, A. Rubin, Y. Svoray, M. Schwartz, and G. Weiss. An efficient shift rule for the prefermax De Bruijn sequence. Discrete Math., 342(1):226–232, 2019.
 [DHS+18] P. B. Dragon, O. I. Hernandez, J. Sawada, A. Williams, and D. Wong. Constructing de Bruijn sequences with colexicographic order: the $k$ary Grandmama sequence. European J. Combin., 72:111, 2018.
 [Etz86] T. Etzion. An algorithm for generating shiftregister cycles. Theoret. Comput. Sci., 44(2):209224, 1986.
 [GSWW20] D. Gabric, J. Sawada, A. Williams, and D. Wong. A successor rule framework for constructing $k$ary de Bruijn sequences and universal cycles. IEEE Transactions on Information Theory, 66(1):679–687, 2020.
 [FM78] H. Fredricksen and J. Maiorana. Necklaces of beads in $k$ colors and $k$ary de Bruijn sequences. Discrete Math., 23(3):207210, 1978.
 [SWW17] J. Sawada, A. Williams, and D. Wong. A simple shift rule for $k$ary de Bruijn sequences. Discrete Math., 340(3):524531, 2017.
 [YD93] J.H. Yang and Z.D. Dai. Construction of $m$ary de Bruijn sequences (extended abstract). In J. Seberry and Y. Zheng, editors, Advances in Cryptology  AUSCRYPT '92, pages 357363, Berlin, Heidelberg, 1993. Springer Berlin Heidelberg.