There are several different ways to obtain a reduction in child support and most of them are based upon a reduction in income by 15% or more or it has been 3 years since the date of the last child support order (COLAs do not count). If you have either of those two situations, the court will hear the case. However, just because they hear it it doesn't mean you win.
The key is to show that the reduction in your income is due to circumstances beyond your control. So if you were fired for cause, or if you quit your job it is unlikely that you will get very far. If on the other hand your employer is reducing salaries, changing your job description, not paying bonuses or overtime has disappeared then you may be able to make out a case. If you have been let go because the company is closing, the company is reducing staff, the job is no longer necessary those would also be examples of events beyond your control.
The next thing you need to do is to keep a good paper trail of your job search. Send out resumes, respond to job adds, consult with a headhunter (if they are available for what you do), show how you have been emailing contacts you have developed, show that you have gone for interviews. The more diligent your job search the more likely you will get a reduction.
Finally, you should show how the job loss or pay reduction has affected your ability to meet your current living expenses. If all of your bills are still paid on time and you have significant savings, you might not get the reduction because the court may thing the only bill you don't want to pay is the one for child support. On the other hand if you are borrowing from 401k, incurring credit card debt, accumulating arrears on bills, then the court will have further evidence of the very real impact on your finances.
Finally, working with an experienced matrimonial attorney will be very helpful as they will be able to sort through the evidence with you and tell you what is missing and what is important.