As you progress, you gradually speed up. However, tripping on an obstacle (e.g. hitting a tile pile or mistiming a gap jump) does slow you down and puts the monster just one misstep away from catching you. But because tripping does not kill you, it can be used to slow yourself down when reacting in time is getting too difficult. After tripping though you accelerate much faster, quickly returning to the speed you were running before you tripped. But once you see that you have regained your lead on the monster (i.e. it is no longer visible at the bottom of your screen), you can trip again, and slow down, without dying.
Dying and saving yourself works similarly to tripping. You do start off slower but quickly regain the speed you lost.
The speed power-up (boost) increases your speed temporarily and gives you full invincibility. You are protected from all falls and obstacles, but are still required to use your tilt controls to gather outer coins (unless you have a magnet). You only slow down relative to the boost speed and return to the speed you were running just before the boost, or to a speed dependent on the slightly increased distance (see below).
A headstart is an extended boost that can be activated at the beginning of your run. You slow to running pace once the headstart distance has been reached. From my experience, the speed you are running after a headstart is slower than how fast you would be running in a flawless run up to that point without a headstart. But like the trips and saves, you might accelerate faster for a short time and eventually run at a speed comparable to a run without a headstart.
My guess is that velocity is a function of distance. If you trip, die and save, or headstart, you reset your velocity but it is quickly brought back to the velocity set for the distance.