The easiest way to make your particles travel along a cure would be to use curve flow in Maya. But it\f you need customized control over that this tutorial will help you.
 
Create curve using the curve tool and also create a particle with emitter.
 
Now select the particle and then the curve and go to the particle menu and click on goal option box and set the weight to 1 and create the goal.

Figure 1: Goal menu

Now go to your per particle attributes of your particles and click on the general tab and add goal U and goal Offset.
 
Now write a small expression in the runtime script of the goal U per particle attribute and given below.
 
particleShape1.goalU += 0.01;

Figure 2. Goal expression

The above line decides actually how fast it should travel over the curve as you increase this value the speed increases.
 
Now if you need a volume offset around the curve use the expression in the creation of the per particle attribute of goal Offset
 
particleShape1.goalOffset = sphrand (<<1,1,1>>);

Figure 3. Goal expression

If you want to reduce the volume size you can reduce the value equally.
 
You will get a similar result as shown below.

Figure 4.  Output