Let $\sigma>0$. Note that $$\kappa(x,\;\cdot\;):=\mathcal N(x,\sigma^2)\;\;\;\text{for }x\in\mathbb R$$ is a Markov kernel on $(\mathbb R,\mathcal B(\mathbb R))$. Let $X_0$ be a $[0,1)$-valued random variable and $X_n$ be a real-valued random variable with $$\operatorname P\left[X_n\in B\mid X_{n-1}\right]=\kappa(X_{n-1},B)\;\;\;\text{almost surely for all }B\in\mathcal B(\mathbb R)\tag1$$ for $n\in\mathbb N$.
While $X_0$ takes values in $[0,1)$, the $X_n$ will leave this interval eventually when they are sampled according to $(1)$. What I would like to do is the following: Assuming $X_{n-1}$ takes values in $[0,1)$, I would like to force $X_n$ to remain in this interval by wrapping around to the other end of the interval if necessary. I guess this could be done by replacing $X_n$ with $X_n-\lfloor X_n\rfloor$. In any case, how do we need to alter the definition of $\kappa$ so that it is a Markov kernel $\tilde\kappa$ on $([0,1),\mathcal B([0,1))$ instead and hence all the $X_n$ take values in $[0,1)$ when they are sampled according to $(1)$ with $\kappa$ replaced by $\tilde\kappa$?
Let $$\iota:\mathbb R\to[0,1)\;,\;\;\;x\mapsto x-\lfloor x\rfloor.$$ Note that $\iota$ is Borel measurable and hence the pushforward measure $\iota_\ast\kappa(x,\;\cdot\;)$ is well-defined for all $x\in\mathbb R$. Thus, $$\tilde\kappa(x,\;\cdot\;):=\iota_\ast\kappa(x,\;\cdot\;)\;\;\;\text{for }x\in[0,1)$$ is a Markov kernel on $([0,1),\mathcal B([0,1))$. Note that $$\iota^{-1}(B)=\biguplus_{k\in\mathbb N}(k+B)\;\;\;\text{for all }B\subseteq[0,1)\tag2$$ and hence $$\tilde\kappa(x,B)=\sum_{k\in\mathbb Z}\kappa(x,k+B)\;\;\;\text{for all }(x,B)\in[0,1)\times\mathcal B([0,1)).\tag3$$ By construction, we easily see that if $X$ and $Y$ are $[0,1)$-valued and real-valued random variables, respectively, with $$(X,Y)\sim\mathcal L(X)\otimes\kappa,\tag3$$ then $\tilde Y:=\iota(Y)$ is a $[0,1)$-valued random variable with $$(X,\tilde Y)\sim\mathcal L(X)\otimes\tilde\kappa.\tag4$$
What I wrote above is correct for a general Markov kernel $\kappa$ on $(\mathbb R,\mathcal B(\mathbb R))$. In the special situation of the question, where the density of $\kappa$ is a "radial" function, we may note that $$\tilde\kappa(x,B)=\sum_{k\in\mathbb Z}\kappa(x,k+B)=\sum_{k\in\mathbb Z}\kappa(x-k,B)=\int\lambda({\rm d}y)\underbrace{\sum_{k\in\mathbb Z}\varphi_{\sigma^2}(y-(x-k))}_{=:\:\psi_{\sigma^2}(y-x)},\tag5$$ where $\varphi_{\sigma^2}$ denotes the density of $\mathcal N(0,\sigma^2)$, for all $(x,B)\in[0,1)\times\mathcal B([0,1))$. So, $\tilde\kappa$ has a radial density $\psi_{\sigma^2}$ as well.