Will something bad happen if different covariates are used to estimate the probability of class membership in multinomial logistic regression? Copying the definition of multinomial logit regression equations from wikipedia, the probability that the $i$-th observation belongs to class $k$ (out of $K$) is:
$$
\begin{align}
\Pr(Y_i=1) &= {\Pr(Y_i=K)}e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i} \\
\Pr(Y_i=2) &= {\Pr(Y_i=K)}e^{\boldsymbol\beta_2 \cdot \mathbf{X}_i} \\
\cdots & \cdots \\
\Pr(Y_i=K-1) &= {\Pr(Y_i=K)}e^{\boldsymbol\beta_{K-1} \cdot \mathbf{X}_i} \\
\end{align}
$$
with
$$\Pr(Y_i=K) = 1 - \sum_{k=1}^{K-1}{\Pr(Y_i=K)}e^{\boldsymbol\beta_k \cdot \mathbf{X}_i}$$
and 
$$\boldsymbol\beta_k \cdot \mathbf{X}_i =  \beta_{0,k} + \beta_{1,k} x_{1,i}  + \beta_{2,k} x_{2,i} + \cdots + \beta_{M,k} x_{M,i}$$
for the vector of covariates $(x_{1,i}, x_{2, i}, \dots, x_{M, i})$ associated with the $i$-th observation.
As written, this seems to imply that in the Wikipedia definition the same set of covariates are used to predict class membership probability. 
What would happen if the we allowed the set of covariates to be different for each regression equation, i.e.:
$$\Pr(Y_i=k) = {\Pr(Y_i=K)}e^{\boldsymbol\beta_{k} \cdot \mathbf{X}_{k,i}}$$
Is this useful? Would it cause problems in model fitting? Or does the original definition already include this case, whereby $\mathbf{X}_i$ is just the union of all $\mathbf{X}_{k,i}$, with some regression coefficients fixed at zero?
EDIT: This is made in response to Tim's answer which I do not get, because of the following example. Let $Y | X$ be categorically distributed with three classes. Then, the generative model is:
$$\begin{split}
\Pr(Y = 3) &= \frac{1}{1 + e^{\alpha x_1} + e^{\beta x_2}}\\
\Pr(Y = 2) &= \frac{e^{\beta x_2}}{1 + e^{\alpha x_1} + e^{\beta x_2}}\\
\Pr(Y = 1) &= \frac{e^{\alpha x_1}}{1 + e^{\alpha x_1} + e^{\beta x_2}}
\end{split}$$
Under what kind of regression modelling framework can the original $\alpha$ and $\beta$ coefficients be recovered?
 A: The multinomial regression model is
$$
\Pr(Y = k \mid \mathbf{X}) =
\frac{
\exp\big( \beta_k \mathbf{X} \big)
}{
1 + \sum_{i=1}^{K-1} \exp\big( \beta_i \mathbf{X} \big)
}
\tag{1}
$$
What you want to estimate is
$$
\Pr(Y = k \mid \mathbf{X}_k) =  
\frac{
\exp\big( \beta_k I(Y = k) \mathbf{X}_k \big)
}{
1 + \sum_{i=1}^{K-1} \exp\big( \beta_i I(Y = i) \mathbf{X}_i \big)
}
\tag{2}
$$
So the first consequence to notice, is that you are not estimating the conditional probability of $Y \mid \mathbf{X}$, at least not in the sense that there is a linear dependence of $Y=k \mid X_k$.
Now, since both terms $\alpha_k$ and $\beta_k \mathbf{X}_k $ are unique for each $k$, then any one of them can, as well, be equal to zero and you would still have the model
$$
\Pr(Y = k \mid \mathbf{X}_k) = \frac{ \exp\big( I(Y =k)\cdot \text{something}_{\,k} \big) }{
1 + \sum_{i=1}^{K-1} \exp\big( I(Y =i)\cdot \text{something}_{\,i} \big)
}
$$
so you model reduces to
$$
\Pr(Y = k) = 
\frac{
\exp\big( \alpha_k \big)
}{
1 + \sum_{i=1}^{K-1} \exp\big( \alpha_i \big)
}
$$
since you are not estimating any conditional probabilities, and for each $k$ the formula needs to return some constant that is equal to $\Pr(Y = k )$.
Saying this otherwise, you would get the same result if $\beta_k I(Y = k) \mathbf{X}_k = \alpha_k$, since $\frac{ \exp( \beta_k I(Y = k) \mathbf{X}_k ) }{ 1 + \sum_{i=1}^{K-1} \beta_i I(Y = i) \mathbf{X}_i } $ needs to evaluate to a single value for each $k$. So $\beta_k I(Y=k) \mathbf{X}_k$ does not let you to estimate any linear dependence of $Y = k \mid \mathbf{X}_k$.
Regarding your example, if you rename $\alpha X_1, \beta X_2$ to $a, b$, then you would clearly see that those values need to be constants, so there is no way how regression parameters would tell you anything meaningful in here. It is as if you had a logistic regression model and used it for data containing only successes (or failures).
