Mutation and crossover operations in discrete differential evolutionary operations?
There isn't a single way in which one can approach a discrete optimization problem using Differential Evolution (DE).

Widespread techniques listed under the Discrete Differential Evolution label aren't DE-specific.

You can allow variables to take values in a continuous range and use penalty functions to enforce integer values:

$$ \bar{f}(w) = f(w) - \sum_i{k_i \cdot (w_i - \operatorname{round}(w_i))^2} $$

$w$ is the vector of parameters (chromosome values), $f: \mathbb R^n \rightarrow \mathbb R$ the basic fitness function (here assuming "greater is better"), $k$ a problem-specific scaling vector, $\bar{f}(\cdot)$ the "penalized" fitness function.

In this way the DE algorithm (DE/rand/1) stays the same:

$$\begin{align}
X_{j,r2}^G - X_{j,r3}^G &amp; = \{2,2,3,0,4,2\} - \{1,2,3,3,0,1\} = \{1,0,0,-3,4,1\} \\
F \cdot (X_{j,r2}^G - X_{j,r3}^G) &amp; = 0.5 \cdot \{1,0,0,-3,4,1\} = \{0.5,0,0,-1.5,2,0.5\} \\
V_{j,i}^{G+1} &amp; = \{4,1,3,2,2,0\} + \{0.5,0,0,-1.5,2,0.5\} = \{4.5,1,3,0.5,4,0.5\}
\end{align}
$$

The trial vector $U$ is obtained via crossover between the donor vector $V_{j,i}^{G+1}$ and a target vector $X$:

$$U_{j,i}^{G+1} = \operatorname{crossover}(V_{j,i}^{G+1}, X_{j,i}^{G})
$$

The target vector is compared with the trial vector and the best one is admitted to the next generation.

This is the recommended procedure with R DEOptim Package (via the optional fnMap parameter).



You can round all the real-valued parameters before evaluating the fitness function:

$$\bar{f}(w) = f(\operatorname{round}(w))$$

(round acts as a repair operator)

This is the technique used by Mathematica's functions NMinimize / NMaximize with the options Method → "DifferentialEvolution" and  Element[w,Integers]



There are also many variations of DE named something-Discrete-DE:


Binary Discrete Differential Evolution: the solution of a problem is presented as a binary string instead of a real-valued vector
Real Value based Discrete Differential Evolution introduces forward/backward transformations to map integer into real number
and viceversa
Exchange based Discrete Differential Evolution: here the crossover operator doesn't change but mutation, being the primary operator acting on
elements of vector in continuous space, is replaced.
...


So you should specify what form of Discrete DE you're interested in for a step by step example.

Meanwhile A Comparative Study of Discrete Differential Evolution on Binary
Constraint  Satisfaction Problems by Qingyun Yang (2008 IEEE Congress on Evolutionary Computation) is a good starting point with many references.