No doubt someone else will suggest a more elegant solution, but try this:
Create a new comp-sized solid. Create a curved mask path in the shape you want your arrow to travel. Apply the Stroke effect and keyframe the End parameter to the speed you want the line to grow.
Add a new Shape Layer and draw your arrow head. Fill it with the same color as you used for the Stroke. Move the anchor point to the center of the arrow head.
Copy your mask path into the Position of the arrow head and set it to Orient Along Path. (You may have to adjust your Rotation to get it pointing in the right direction.)
Select all your Position keyframes on the arrow head , hold down Alt/Option and pull the last keyframe in to match the last keyframe on your Stroke effect. The arrow head should now move on and appear to leave a path trail behind it.
I'm sure there's a much cooler way involving expressions and Slider controls, but this should at least get you going.