Continuous (yeast) culture system satisfying multiple criteria
A chemostat should satisfy your criteria. 
The cell mass balance around a chemostat is the following (assuming no biomass in the feed):
$$F-FX+V_RX\mu_g-V_Rk_dX=V_R\frac{dX}{dt}$$

At steady state there is no change in parameters with respect to time so we set the derivative equal to zero. If we also assume that the death rate ($k_d$) is negligible (for example cells in exponential/log phase) and let:
$$D=\frac{F}{V_R}$$

Where $D$ represents the dilution rate (how quickly the volume in the tank is being replenished). Then rearranging the above equation you get:
$$D=\mu_g$$

This says that the growth rate will be equal to the dilution rate. In other words you can set how fast the cells grow by modifying the flow rate and/or volume of the tank. But there is a limit. Cells grow as fast as they can use the limiting nutrient ($S$) and this relationship is often modeled as:
$$\mu_g=\frac{\mu_mS}{K_s+S}$$
Where $\mu_m$ and $K_s$ are unique parameters for every cell type. $S$ is the concentration of the limiting substrate in the tank (which is also equal to the concentration of the substrate in the outlet stream). The maximum growth rate is $\mu_m$ and if the dilution rate is higher then this value, the cells get washed out of the chemostat. This can be troublesome if you need a flow rate higher than $\mu_m$. In this case a recycle is useful.

To answer your question in the comments regarding medium composition. You probably want to operate at steady-state. Steady state means that there is no change of parameters with respect to time. So if you write the substrate mass balance:

$$FS_0-FS-V_RX\mu_g\frac{1}{Y_{\frac{X}{S}}}--V_RXq_p\frac{1}{Y_{\frac{P}{S}}}=V_R\frac{dS}{dt}$$

And set the derivative to 0. Then you will find the outlet substrate concentration is a function of other constants (biomass, volume, product yield, etc.). You're right in that some substrates get used more than others, but when steady state is reached. They stop changing and the medium composition stays constant in the vessel.