I'll give this a bash.
http://en.wikipedia.org/wiki/Schwarzschild_metric" represents curved space-time in a vacuum in the following form (assuming that d\theta and d\phi equal zero)-
ds^2 = \left(1-\frac{2M}{r} \right)dt^2 -\left(1-\frac{2M}{r}\right)^{-1}dr^2
where M=Gm/c^2 (often referred to as the gravitational radius) where G is the gravitational constant, m is the mass of the object, c is the speed of light, dt and dr are change in time and distance respectively and r is variable, reducing the closer you get to an object of mass.
proper time would be represented by-
dt_{shell}=\left(1-\frac{2M}{r} \right)^{1/2}dt
where dtshell represents the time dilation at a specific (coordinate) radius.
and proper distance would be represented by-
dr_{shell}=\left(1-\frac{2M}{r} \right)^{-1/2}dr
where drshell represents the distance inflation at a specific (coordinate) radius.
Velocity is normally expressed as v=m/s (distance/time) and the velocity induced by spacetime curvature could be expressed as-
v_{shell}=\frac {dr_{shell}}{dt_{shell}}
If we substitute the above equations for dtshell and drshell, we get the following equation-
v_{shell}=-\left(\frac{2M}{r}\right)^{1/2}
negative because the object is moving away from the observer towards the source
Multiply by c for m/s. Objects take the shortest path through spacetime so if it feels a time dilation, no matter how slight, to one side of it, it will tend towards the source of the time dilation (or curvature). v increases as r reduces which is in some way analogues with Newton's equation for gravity g=Gm/r^2 (technically, gravity is g=dr_{shell}\cdot Gm/r^2 but drshell is normally only included when calculating gravity for ultra-compact objects such as neutron stars and black holes and normally ignored for less dense objects such as planets). If you apply the above to Earth say, you'll notice that vshell for a free-falling object at the Earth's surface is -11.2 km/s which is the negative of the escape velocity expressed as v_{esc}=\sqrt(2Gm/r) which means that an object that fell from rest at infinity would hit the Earth's atmosphere at ~11.2 km/s, 'moving along the curved spacetime geodesic' caused by Earth's mass. (Note: The above velocity represents an object free-falling from infinity only and doesn't account for an object falling from rest at a specific radius) In Minkowski space (i.e. flat space) the velocity (which would be the result of work done by an external source) would remain constant and there wouldn't be any acceleration.
It's also worth noting that for an object free-falling from infinity, E/m=1, the energy required for an object to remain stationary in curved Schwarzschild space is-
\frac{E_{shell}}{M}=\left(1-\frac{2M}{r}\right)^{-1/2}
so unless there's an input of energy, the object has to move along the curved spacetime geodesic.
