I agree with most of what's been said so far. If I may offer my insight into chain (chase) punching...
1. During sparring we're taught to only CP when we're in close range. This seems obvious but some people get into a bad habit of trying to force a CP when they're out of range and just blasting air without following up with a more long range strike (ie. arrow punch or front kick).
2. When you are in good range to use CPs, mix it up with both leveled and low chain punches (sinking w/ your horse). This can further disrupt your opponnent not knowing where to cover high or low when your blasting with your punches and then follow up with a different strike as I described in 1.
3. For us, CP is never the initial attack or I should say counter-attack. It's usually the next technique in a series of combinations of other techniques.
Keep in mind it's all tactics when using CPs as with any techniques. Also maintain the balance of keeping it simple but also giving your opponents different looks.