Event-based rather than flow-based systems? This is a very vague question, so I don’t know if people will understand it (I’m not even sure I understand the question myself), but here it goes:
Dynamical systems are formulated (in continuous time) as
$$\begin {align}
&\dot x_1=f_1(x_1,...,x_n)\\
&...\\
&\dot x_n=f_n(x_1,...,x_n)
\end {align}$$
These are flows over time. In other words, we have a stock for each variable, and a specified flow at each point in time, from which we will deduce the stock over time. 
Situations in applied science that are captured well by this system are for example:


*

*There is an amount of water $x$ in a cup, and we know that every second, an amount $a$ drips through a hole, so that $\dot x=a$.

*An object is moving with $x$ meters/s towards the earth, and we know that every second, $x$ increases by $9.8\,\mathrm{m}$.


We can of course do this in discrete time as well, which doesn’t fundamentally change the idea:
$$\begin {align}
&\Delta x_1=f_1(x_1,...,x_n)\\
&...\\
&\Delta x_n=f_n(x_1,...,x_n)
\end {align}$$

However, sometimes our logical analysis in an applied setting doesn’t start with knowledge of a flow per unit time, but with knowledge of events. For example, in biology, a birth can occur, which would cause various things to happen (among other things, an increase of the population by $1$). 
As we do in most of science, we also generally implement births in population models as a flow. Instead of starting with a formal definition of a birth as an event, we simply state that in a given time interval, the population increases. In other words, the analysis starts with the variable population, and with a formula denoting its change over time. In fact the whole concept of a birth, being an event, never enters the mathematical model, and is only a kind of informal interpretation that we stick to the model.
I am wondering whether this informal leap from the event of a birth to a flow can somehow be formalized.
So my questions are:


*

*Is there a (sub)branch of mathematics that starts its analysis with a formalization of an event, rather than of a flow?

*Is there any mathematical analysis of events at all?
I tend to think of events as function calls in the programming sense, i.e., when a birth occurs, a “birth” function is called and this causes various variables to change, and possibly other events.
 A: 
As we do in most of science, we also generally implement births in population models as a flow. Instead of starting with a formal definition of a birth as an event, we simply state that in a given time interval, the population increases. 

This is not generally true. Unfortunately, in many cases we teach our students this way, carefully hiding the underlying discrete nature of many processes. In particular, if you speak about biological growth models, there is always a discrete stochastic process underlying it, and, ideally, one needs to formulate any model starting with such process. The truth is that, however, we can build a transparent and simple mathematical theory only in linear case. In this case, for instance, the standard Maltusian growth
$$
\dot N=\mu N
$$
naturally appears as the equation for the mean population number $N(t)=\rm E[X(t)]$, where $X$ is described by a simple birth and death process.
In the nonlinear case the mathematics becomes much more demanding, and although we often teach our students that our continuous model describe some expectations of the quantities of interest, this in not true in general. To get more ideas I would recommend you to give  a look at


*

*From individuals to epidemics

*Interacting Particle Systems
which, together with the references therein, give a general answer to yout two questions.
