**Proposition 1:** The number of integer solutions of the equation
$$
\sum_{i=1}^{k}x_i = N
$$
where $x_i\geq n_i$ for $i=1,\ldots,k$, is given by
$$
{\small
\binom{N+k-1-n_1-n_2-...-n_k}{k-1}
}
$$
if the upper index is non-negative and zero otherwise.

In the formula above, $\binom{.}{.}$ stands for the generalized binomial coefficients.

Now, to tackle the problem as stated, you need to apply Proposition 1 and invoke the inclusion-exclusion principle, in the following sense:

For $i=1,...,k$, set as

$q_i$: the property of a solution of Proposition 1, to satisfy the condition
$$
x_i> m_i \Leftrightarrow x_i\geq m_i+1
$$

If we denote:

- $N(q_i)$, the number of solutions (provided by Prop. 1) satisfying property $q_i$,
- $N(q_i q_j)$, the number of solutions (provided by Prop. 1) satisfying both properties $q_i$, $q_j$,

... and generally:

- $N(q_{i_1}q_{i_2}... q_{i_s})$, the number of solutions (provided by Prop. 1) satisfying all properties $q_{i_1}$, $q_{i_2}$, ..., $q_{i_s}$,

then we get -applying Prop. 1- that:
$$
{\small
N(q_1)=\binom{N+(k-1)-1-m_1-n_2-...-n_k}{k-1} \ \ \ or \ \ \ N(q_1)=0
}
$$

$$
{\small
N(q_2 q_3)=\binom{N+(k-2)-1-n_1-m_2-m_3-n_4-...-n_k}{k-1}
\ \ \ or \ \ \ N(q_2 q_3)=0}
$$
... and generally:
$$
{\small
N(q_{i_1}q_{i_2}... q_{i_s})=\binom{N+(k-s)-1-\sum_{i\notin I} n_i-\sum_{i\in I} m_i}{k-1} \ \ \ or \ \ \ N(q_{i_1}q_{i_2}... q_{i_s})=0
}
$$
where $s$ is the number of properties, $I=\{i_1, i_2, ..., i_s\}\subseteq \{1,2,...,k\}$ and the $N(..)$ function takes zero values whenever the upper index becomes negative.

Now all you need to do to obtain a compact formula for the number of solutions satisfying your constraints, is to apply the inclusion-exclusion principle to determine the number of solutions produced by Proposition 1, **which have none** of the properties $q_i$ for $i=1,2,...,k$.

This is given by

$$
\binom{N+k-1-\sum n_i}{k-1}-\sum_{i=1}^{k} N(q_i)+\sum_{k\ \geq j > i\geq 1} N(q_i q_j)-...+ \\ +(-1)^s\sum_{k\ \geq i_s>...>i_1\geq 1} N(q_{i_1}q_{i_2}... q_{i_s})+.... +(-1)^k N(q_{1}q_{2}... q_{k})
$$

where in the above formula $s$ is the number of properties and
$$
\sum_{k\ \geq j > i \geq 1}=\sum_{i=1}^{k-1}\sum_{j=i+1}^{k}
$$
... etc.

**Example:** As an example of application of the previous method, consider the following special case of the OP:

Find the number of (positive) integer solutions of the equation
$$\sum_{i=1}^{k}x_i = N$$ for some positive integer $N$, given the constraints $1\leq x_i\leq \alpha$ for $i=1,\ldots,k$

The method described above gives:
$$
{\small
\binom{N-1}{k-1}+\binom{k}{1}\binom{N-\alpha-1}{k-1}+\binom{k}{2}\binom{N-2\alpha-1}{k-1}+\binom{k}{3}\binom{N-3\alpha-1}{k-1}+\cdots
}
$$
where $\binom{..}{..}$ stands for the generalized binomial coefficients and the summation halts when zero terms appear.