How to track asteroids or NEOs? Fast-moving objects near our earth have different paths in the sky. They are different with stars far away. 

For telescopes, when we do not know an object's orbit, catching up with its orbit and not losing it is a problem ? How to figure out a new object's orbit parameters quickly before we lose it? For an old object with an irregular orbit, can a telescope track it? 

The typical speed in the asteroid belt may be smaller than 1"/min? But there may be faster objects. Then the paths are not so simple and regular.
Specialty hardware will be able to do it, as might clever programming if you have a commercial, computerized mount and feel like hacking away. Either way, an ordinary mount will be able to follow targets in the asteroid belt easily -- their motion across our sky isn't actually very fast. What I'm about to write really only applies to NEOs.

With NEOs, the problem isn't just the apparent speed of the objects across our sky, it's that the objects are usually in irregular orbits while moving at that speed. This is in addition to the apparent motion due to the rotation of the earth.

To track these objects you'd have to project the position of the object onto our sky, using the object's ephemeris data as your starting point, then add the apparent motion due to the rotation of the earth. That position could be calculated by the mount or calculated by a computer tethered to it, which would send updated pointing directions to the mount. The specifics on how to do this depend on your hardware. It's not a small project.