Flow and Time derivative? If $a_{t}$ is the asset at time $t$, $c_{t}$ the consumption at time $t$, and $r$ the interest rate, then $a_{t+1} - a_{t} = ra_{t} - c_{t}$ is the flow of $a$. But it seems that in macroeconomics people just write $\dot{a}_{t} = ra_{t}-c_{t}$ instead in continuous time setting. I do not see the passage from $a_{t+1}-a_{t}$ to $\dot{a}_{t}$ in general. I can justify it in some special cases. For instance, if $a_{t+1}-a_{t} = r$ for some $r$, all $t$, then $a_{t+h}-a_{t} = hr$, so taking limit of the difference quotient leads us to $\dot{a}_{t} = r$. But this depends on the fact that the right-hand-side term is linear in $t$.
I am unsure how economists deal with the transition from continuous time to discrete time. My background is in systems engineering/applied mathematics, and the economics literature I am familiar with is either discrete time or continuous time. Unfortunately, I have no good references on the conversions from discrete to continuous time from the systems engineering literature (got rid of the textbooks).

The key in this case is that the “a” coefficient in these systems do not necessarily correspond to the same number. We need to adjust it so that behaviour lines up.

If we are transitioning from continuous time to discrete time, we effectively are sampling the continuous time system. The frequency we sample at matters.

This is easy to see. Imagine that a continuous variable grows by 1% in six months. If the discrete time step is six months, the variable grows by 1.01 each period. However, if we our sample is annual, we get a transition growth step of $(1.01)^2$ per period. Essentially, the formula to convert the growth coefficient is similar to changing interest rate quote conventions from continuous compounding to a bond yield equivalent.

Although a continuously growing variable (like a compunding deposit) is relatively straightfoward to deal with, something like consumption series is more difficult to specify. If we want the discrete time to correspond to the sampled values of the continuous time version, we need to pin down the continuous time dynamics in some fashion. Is it constant during the time interval, or is it also growing in an exponential fashion (which would be the case if it were an endogenous variable inside a linear model)?

Going from discrete time to continuous time is more awkward. A continuous time system contains much more information than a discrete time system. There is an infinite number of continuous time systems that can generate the same discrete time system. We need a machanism to pin down the dynamics of the continuous time version, but the decision can be viewed as arbitrary. This is a topic that is not heavily studied in systems engineering, as physical systems are already truly operating in continuous time.

A simple economic example of the loss of information is as follows. Imagine that we have a quarterly model frequency, but a household receive a cash flow at some point during the quarter. Interest compounds daily (like a bank deposit). The final amount held at the end of the quarter depends upon the length of the holding period, which could be from 0-3 months in length. Since the discrete time model has no notion of times in between the quarterly sample periods, it has no way of modelling this timing effect that is a property of the continuous time model. This means that a discrete time model is likely only going to approximate a continuous time model.