Question: Most importantly, what must change in order for the falling object to change its speed? Is it the distance to the centre of the planet? If you pull the earth away from the object as the object falls, will the object slow down or will it keep accelerating?

Answer: 
As long as there's a non-zero net force acting on the object, it will have a non-zero acceleration and therefore it will continuously change its velocity: $$\vec{F} = m\vec{a}.$$

In the case of gravity, the force is inversely proportional to the distance between the objects squared: $$\vec{F}\_G = G\frac{m\_1m\_2}{r^2}\frac{\vec{r}}{r},$$ where $\vec{r}$ is the vector connecting the two objects and $r=|\vec{r}|$ its length. So the closer the objects are, the stronger the force or -equivalently- the acceleration. Notice that the acceleration is only zero if the objects are infinitely far apart. (I'm assuming no drag, let's only consider the gravitational force for simplicity)

If you pull the earth in the same direction the object is falling so that you maintain the same distance $r$ at all times the object will just keep falling with a constant acceleration. If you pull it faster, the distance will increase and the acceleration will therefore decrease, meaning the velocity of the object will increase more slowly than before. But it will never decrease. Pulling the earth more slowly will only decrease how much the acceleration would increase if you hadn't pulled, so again the velocity keeps increasing.

So to summarize, the object's velocity will always increase, unless you can get the distance to infinity, which should only take you about - an infinite amount of time. And even then you can only get to a constant velocity, never a decreasing one. You need a repulsive force for that (or additional attractive forces on the other side of the object).
> 
> Most importantly, what must change in order for the falling object to change its speed?
> 
> 
> 

Obviously, there's nothing necessary for that. Ignoring ***drag***, the velocity of the object is increasing for sure. Only the acceleration remains constant. Taking *air effects* into account, the object attains the terminal velocity (based on fluid), its $v$ remains constant & $a$ becomes zero. If you're concerned about changing its velocity now, then you either apply another force or *remove* the fluid...

> 
> Is it the distance to the center of Earth?
> 
> 
> 

Well, quite... Once the distance to the center increases, the gravitational pull exerted on the object decreases - leading to less acceleration. Hence, velocity changes *indeed*...

> 
> If you pull the earth away...
> 
> 
> 

The acceleration on the object decreases slowly. If another body attracts it with some comparable force, your object stops accelerating towards it. But, never quite reaches stationary state...

---

Whenever a force is exerted on an object, there's a change in its velocity as a function of time. It's old-school. Check out [Newton's laws](http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion#Newton.27s_second_law) and finally substitute gravity and you're done...