Are RNN or LSTM appropriate Neural Networks approaches for multivariate time-series regression?
Here is a really good source to begin multivariate time-series forecasting in Keras using LSTMs. 


  I aim to train my model/regression on an interval $[t_0,t_n]$ and then be able to apply it on a larger different interval.


There's no harm in this as long as you perform the right kind of multi-step forecasting. If your problem requires you to train on $[t_0, t_n], \text{for some } n &lt; 100$ and produce $y_1(t_{n+1}), y_2(t_{n+1})$ as outputs then APPLY it on $[t_n, t_n+100]$ then there will be issues in implementation, as most ML models will require you to provide the same shape as you did when you were training your model. Here's when sliding across time-series will help you. Simply put, 

$$
\begin{align}
\text{Train on: }&amp; [t_0, t_n] &amp;\text{ Output: }&amp; y_1(t_{n+1}), y_2(t_{n+1}) \\
\text{Predict on: }&amp; [t_1, t_{n+1}] &amp;\text{ Output: }&amp; y_1(t_{n+2}), y_2(t_{n+2})\\
\text{Predict on: }&amp; [t_2, t_{n+2}] &amp;\text{ Output: }&amp; y_1(t_{n+3}), y_2(t_{n+3})\\
&amp; \vdots &amp; &amp; \vdots\\
\end{align}
$$


  I know that relation between my Y and X are non-linear, but also that in need to take into account lag or inertia. For example, Y1(t) is dependent of X1([t−dt1,t]) and X2([t−dt2,t]). All dtn are different, but I have an approximate idea of how 'far' I need to reach.


Well, you can model your time-series data as $X_1(t-k),\cdots,X_1(t), X_2(t-d),\cdots,X_2(t)$ as $X$ (input) and $X_1(t+1), X_2(t+1)$ as $y$ (target), where $k$ and $d$ refer to different lags for each variable.