I reproduced the canonical difference-in-differences (DiD) equation from your question below:
$$
y_{it} = \gamma Treat_{i} + \gamma Post_{t} + \delta(Treat_{i} \times Post_{t}) + \epsilon_{it},
$$
where, for example, we observe universities $i$ in years $t$. The subscript $i$ usually represents an aggregate unit (e.g., individuals, universities, counties, states, countries, etc.); some of these units receive some treatment/intervention, while others do not. The generalization of this equation, which allows for staggered treatment adoption, regresses your outcome $y$ (i.e., academic score) on a treatment indicator, and dummies for each university and each year. The following specification is a 'generalized' DiD equation which takes the following form:
$$
\text{Score}_{it} = \gamma_{i} + \lambda_{t} + \delta \text{President}_{it} + \epsilon_{it},
$$
where $\text{President}_{it}$ is equal to 1 for universities receiving the new president/rector—and only during years $t$ when the president/rector is actually serving in this position. $\gamma_{i}$ denotes university (unit) fixed effects; $\lambda_{t}$ denotes year (time) fixed effects. Your model will result in 572 separate “university” effects and 9 separate “year” effects. This may seem unwieldy, in practice, but functions exist in most software packages (e.g., R/Stata) to avoid extraneous output. Note, these fixed effects replace $Treat_{i}$ and $Post_{t}$, respectively, in the former equation. Your causal estimand of interest should be $\delta$.
The treatment variable $\text{President}_{it}$ is your interaction term $(Treat_{i} \times Post_{t})$. In the more general DiD setting, though, $Post_{t}$ is not well defined. Instead of specifying this interaction manually, we explicitly code a treatment dummy to reflect early/late adopter universities. Again, $\text{President}_{it}$ should be equal to 1 in only those university-year combinations when the treatment (i.e., president/rector assumes position) is in effect, 0 otherwise. For universities never receiving a new president/rector, it should be coded 0 for the entire observation period.
I encourage you to review this answer which details the coding of the treatment dummy in greater detail.
