I will focus on ARMAX versus VAR. I am not quite sure what a dynamic regression is. (I have seen a few different interpretations. Funnily, there are textbooks and lecture notes with chapters called "Dynamic regression" that do not really delimit this class of models. Also, Rob J. Hyndman notes in his blog post ["The ARIMAX model muddle"][1] that *different books use that term for different models*).

An **ARMAX** model has the form

$$ y_t = \beta x_t + \varphi_1 y_{t-1} + \dotsc + \varphi_p y_{t-p} + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \dotsc + \theta_q \varepsilon_{t-q}. $$

1. The dependent variable is a *univariate* time series. 
2. The model *cannot* be used for forecasting $y_{t+h}$ unless one has the future values of the independent variable $x_{t+h}$ available, or has a separate model for predicting $x_{t+h}$.
3. The model is estimated using maximum likelihood (*slow*), often using a state space representation.

A **VAR** model has the form

$$ z_t = \varphi_1 z_{t-1} + \dotsc + \varphi_p z_{t-p} + \varepsilon_t $$

where $z$ is a vector; for example, $z=(y,x)'$.

1. The dependent variable is a *multivariate* time series. 
2. The model *can* be used for forecasting all components of $z_{t+h}$, e.g. for $z=(y,x)'$. Given data up to and including time $t$, forecasts for time $t+1$ are straightforward to obtain; forecasts for $t+h$ where $h>1$ can be obtained iteratively.
3. The model can be estimated using OLS or GLS (*fast*).

The two could be combined to obtain the **VARIMAX** model that has a *multivariate* dependent variable, *does* allow for forecasting of all of its components but also takes a *long time* to estimate and may often run into *convergence problems*.

[1]: http://robjhyndman.com/hyndsight/arimax/