Does *advection* describe the change of density of massless infinitesimal tiny *thingies* injected into a fluid? I'm considering an incompressible Newtonian fluid with uniform density and try to figure out what's meant by the term advection.
Let $\Omega_0\subseteq\mathbb R^d$ be an (infinitesimal small) bounded domain occupied by the fluid at time $0$. If I'm not terribly wrong, advection is all about measuring the change of a thing injected into the fluid. Let's denote the (unitless) amount of that thing injected at $x\in\Omega_0$ at time $t\ge 0$ by $c(x,t)$.
If I'm not wrong, we want to find an expression for $$\frac\partial{\partial t}c(x,t)$$ in terms of the velocity field $u:\Omega_0\times[0,\infty)\to\mathbb R^d$ (and the pressure $p:\Omega_0\times[0,\infty)\to\mathbb R$?) of the fluid.

Is that the case or am I wrong? And whatever the case is: How can we derive the advection equation?

I don't understand the example of "ink dumped into a river". What's the thing in that case? Ink itself is obviously not a scalar quantity. And, even worse, should the process of pouring ink into water not effect the motion of the water? Is can't see that this obvious fact is taken into account.
Or is everything less complicated an the thing is something which doesn't effect the motion of the fluid?

I could imagine that advection just describes the change of density of massless infinitesimal tiny thingies injected into the fluid, but I'm unsure whether I'm wrong or not.

 A: For some background, consider my answer to your question on a similar topic over at ME which i guess is the 'inspiration' for this question.

If I'm not terribly wrong, advection is all about measuring the change of a thing injected into the fluid. Is that the case or am I wrong? And whatever the case is: How can we derive the advection equation?

To be blunt, this is incorrect; Advection is a mechanism with which a quantity (mass, energy, momentum, entropy, etc.) is transported by a velocity field. This is in contrast with diffusion which is a mechanism with which the quantity is transported by spatial gradients of that quantity. The combined effects of advection and diffusion is known as convection. Sometimes you will see convection and advection refering to the same thing; this is technically wrong.
There is no such thing as 'deriving the advection equation', because it is a constitutive equation similar to Fick's law. What i think you mean instead is how to derive the 'advection-diffusion' equation which is typically a differential equation relating the accumulation of a quantity to the in- and outflux of the quantity at the boundaries and production or consumption of the quantity. For the derivation see my your question over at ME.

I could imagine that advection just describes the change of density of massless infinitesimal tiny thingies injected into the fluid, but I'm unsure whether I'm wrong or not.

The 'advection-diffusion' equation models the evolution of the density of massless infiniteesimal tiny thingies injected into the fluid. Advection would simply predict how the tiny thingies are transported by the fluid flow. Note that the 'advection-diffusion' is a one-way coupled model with the fluid flow, i.e. the massless tiny thingies follow the fluid flow, but they do not affect the hydrodynamics in any way. So in your comment on the ink influencing the flow (while potentially realistic), the ink could not be modelled by an 'advection-diffusion' equation; instead a more complex two-phase eulerian approach with the Navier-Stokes equations would be suitable.

Does advection describe the change of density of massless infinitesimal tiny thingies injected into a fluid?

In conclusion, no. but the 'advection-diffusion' equation will.
