Say that $D_{it}\in\{0,1,\ldots,K\}$ is a categorical variable denoting whether unit $i$ at time $t$ got treatment $k=0,1,\ldots,K,$ where 0 denotes the absence of treatment (control group).
The answer to your question is yes, if by difference-in-differences you mean estimating via least squares the regression
\begin{align*}
Y_{it} = \alpha_i + \lambda_t + \sum_{\ell=1}^K\tau_\ell D^{\ell}_{it} + \epsilon_{it},\quad i\in[N],t\in[T]
\end{align*}
where $Y_{it}$ is the outcome of interest, $\alpha_i$ and $\lambda_t$ are unit and time fixed effects, and $D_{it}^\ell=\mathbb{1}(D_{it}=\ell)$. You can add control variables if you feel you need to.
The answer to your question is it depends, if on top of the statistical model described above you want to add causal conclusions on the effect of the various treatments on your outcome of interest. If this is the case, there is no one-fits-all strategy and it is really much context-dependent. The answer would depend on how you define the various treatment levels, how your sample looks like, what are your units, how big (i.e., aggregate) your crisis shocks are and so on.
However, this does not mean that there is no hope. The causal inference literature has been recently flooded with papers on the topic. Here a (definitely non-exhaustive) list of some relevant papers for your case:
- Goldsmith-Pinkham, Hull, and Kolesar (2023)
- de Chaisemartin and D'Haultfoeuille (2022)
- Sun and Abraham (2022)
PS (feel free to ignore this): My hunch is that, even ignoring all the problems mentioned by the papers above, in a setting like the one you describe, it is really unlikely for the parallel trend assumption to hold. Your treatment assignment process -aka the fact that a crisis happens- is very much likely to be endogenous. In other words, certain units might have higher chances of experiencing a crisis and this might be related to their potential outcomes in ways that violate parallel trends. Other two subtler assumptions, quite often ignored in DiD settings, that are likely to be violated are:
- the SUTVA (Stable Unit Treatment Value Assumption) would be violated if your shocks are too big/aggregate. The SUTVA require unit $i$'s treatment status to not affect unit $j$'s ($i\neq j$) potential outcomes. This is extremely unlikely if your crisis are systemic or at the local labor market level for example. Much more likely if you are working with small firms, for example, and your definition of crisis is related, say, to the outstanding debt of the company. Again context matters.
- The no anticipation assumption is violated if the treatment is endogenous or somehow foreseeable by your units. Again, this is very unlikely to hold if you are studying some sluggish/slow-moving phenomenon.
