Multiprocess models I would like to  use multi-process models. in particular, I would like to apply this model: Kulu, Hill. "Migration and fertility: Competing hypotheses re-examined." European Journal of Population/Revue européenne de Démographie.
In the fertility equations I don't understand how the estimation procedure differs between the two models (conceptually I get the difference but I am rather confused when it comes to the actual methodological application). In particular, to let the errors vary across the three migration equation is the estimation procedure extended in a discrete-time setting?
Additionally, i don´t understand whether the author distinguishes between a first birth after first migration and first birth after second migration or whether he is just considering first birth after whatever migration.
Can someone help me to better understand? Textbooks about the topic are welcome
thank you! 
 A: These are multilevel multi-state discrete time competing risks models. 
When there are $R_{i}$ types of transition, $ h^{r_{i}}_{tijk}$ is the hazard of transition $r\;  (r_{i} = 1,...,R_{i})$ happening during episode $j$ of individual $k$ who is in state $i$ at time $t$, and $h^{0}_{tijk}$ is the hazard of no transition happening. The effect of duration is represented by $\alpha^{(r_{i})^{T}}_{i} z^{(r_{i})}_{ti}$, with $ z^{(r_{i})}_{ti}$ representing time $t$ for state $i$ (time is state specific). The covariates and their effects are shown as $\beta^{(r_{i})^{T}}_{i}x^{(r_{i})}_{tijk}$, and may vary according time, episode, state, and individual. Lastly, $u^{(r_{i})}_{ik}$ are individual random effects that are state specific. 
$
\log \left( \frac{h^{r_{i}}_{tijk}}{h^{0}_{tijk}} \right) = \alpha^{(r_{i})^{T}}_{i} z^{(r_{i})}_{ti} + \beta^{(r_{i})^{T}}_{i} x^{(r_{i})}_{tijk}  + u^{(r_{i})}_{ik}, r_{i} = 1,...,R_{i}; i=1,...,s 
$  (1)
Regarding your questions:

In particular, to let the errors vary across the three migration equation is the estimation procedure extended in a discrete-time setting?

These models estimate several equations (simultaneous equations) in a multilevel setup, each equation with its own individual random effects $ u^{(r_{i})}_{ik}$. Although there is a discrete time setting, it is the multilevel setup that lets them have different "errors" (individual random effects) for each transition.

Additionally, i don´t understand whether the author distinguishes between a first birth after first migration and first birth after second migration or whether he is just considering first birth after whatever migration.

It seems that he distinguishes between first, second, and third births in general.

Textbooks about the topic are welcome

This book has a chapter on this type of model, albeit using outdated software. The articles in the references below introduce the methods and applications, and one of them was co-authored by the same person from the article you read. 
References
Steele, F., Goldstein, H., & Browne, W. (2004). A general multilevel multistate competing risks model for event history data, with an application to a study of contraceptive use dynamics. Statistical Modelling, 4(2), 145–159.
Kulu, H., & Steele, F. (2013). Interrelationships Between Childbearing and Housing Transitions in the Family Life Course. Demography, 50(5), 1687–1714.
