What is the name of this kind of games?

In game theory, suppose we have a set of players $\mathcal{N}=\{1, 2, \ldots, n\}$, a set of actions $\mathcal{A}_i$ of player $i\in\mathcal{N}$, and a payoff function $u_i$ of player $i\in\mathcal{N}$ defined as $u_i:\mathcal{A}_1\times\mathcal{A}_2\times\ldots\mathcal{A}_n\mapsto\mathbb{R}$.

Further, suppose that the action set of a player $i\in\mathcal{N}$ changes every time $i$ chooses an action.

Example: For example, say $\mathcal{A}_i=\mathcal{A}\,\forall\,i$. Now, if player $i$ chooses $a_i\in\mathcal{A}$, then player $j\neq i$ can choose an action $a_j$ from $\mathcal{A}\backslash\{a_i\}$. If player $j$ does choose $a_j$, then player $k\neq i$ and $k\neq j$ can choose an action $a_k$ from $\mathcal{A}\backslash\{a_i, a_j\}$ and so on.

How do we define such games? Is there any reference to such games? Is there anyway to model these kind of games?

You might ask why this is not really in game theory literature, but rather assignment / matching. I think the reason is the game you describe is not interesting if $u_i = f(a_i)$, because each agent will just pick her preferred remaining action. If $u_i = f(a_i,a_{-i})$, then it's potentially an interesting game. I haven't seen that case analyzed. In fact, I've never seen the game you describe analyzed in game theory stuff.