Why does a heatsink level out at "a temperature" Lets assume I'm talking about a computer-type aluminium heatsink, which has a constant thermal load into its base. 
Over this heatsink moves air at a constant rate, and a constant temperature.
Why is it that a heatsink will apparently come up to a temperature and sit there?
 A: The hotter the heat sink gets, the more quickly it transfers heat to its surroundings. Put mathematically, the heat flux is proportional to the temperature gradient,
$$ q=-k\nabla T$$
where $k$ is the thermal conductivity (which does indeed depend on temperature in most systems, but that's irrelevant here).
So as the temperature increases, and the heat flux also increases, eventually the heat flux out will match the heat flux into the sink, at which point you have equilibrium and the temperature remains constant.
To be specific, suppose the heat power into the heat sink is $P_\text{in}$.
Suppose also that the air temperature is $T_\text{air}$.
Ignore the fact that the air is flowing and just instead assume that the air has a fixed temperature $T_\text{air}$.
Then the heat power flowing out of the heat sink is $P_\text{out} = k (T_\text{heat sink} - T_\text{air})$.
We can just check to see if there is a steady state considtion, i.e. does $P_\text{in}$ ever equal $P_\text{out}$?
\begin{align}
P_\text{in} &\stackrel{?}{=} P_\text{out} \\
&= k(T_\text{heat sink} - T_\text{air}) \\
T_\text{heat sink} &= \frac{P_\text{in}}{k} + T_\text{air} \, .
\end{align}
So there you see it: when the heat sink temperature gets up to $P_\text{in}/k + T_\text{air}$ then the system is in a steady state.
The power into the heat sink is exactly balanced by the power out, so its temperature stays constant.
