# Alternatives to Dickey-Fuller test

I'm studying unit root tests and therefore the Dickey Fuller test and I can't seem to figure the following out.

Is it correctly understood that the Dickey Fuller test (with drift and constant) is designed such that under the alternative the data generating process is

$$y_t=c+\lambda t+\alpha y_{t-1}+\varepsilon_t?$$

If yes then i get we want the augmented Dickey Fuller to have a richer class of data generating processes as alternative so we consider the auxilliary regression

$$y_t=c+\lambda t+\alpha y_{t-1}+\sum_{i=1}^k \beta_i \Delta y_{t-1} +\varepsilon_t$$

where $k$ is chosen such that the $\varepsilon$'s are white noise. Which now allows for alternatives $AR(k)$ models (is that correct?). Okay, now the way the test was presented to me the alternative is that the process is stationary, but how does the test perform if i.e. the actual data generating process is say $MA(q)$ or $ARMA(p,q)$? Do we somehow beforehand narrow $y$ down to being either $AR$ or a unit root process?