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The process is to simply contact the credit bureau and ask if they would remove a negative item sooner than it would fall off on its own. The EE is a request and totally at the discretion of the CRA. They have no obligation to do so, and every EE should be considered an act of generosity on their part.
The typical periods they will will grant them are:
TU - 6 months early
EX - 3 months early
EQ - 1 month early
For example, if you have something that will automatically be removed in September, you probably would be able to get an EE in March, June, and August.
The EE is different than a dispute, but each CRA handles it differently. You generally need to ask for a supervisor when calling. But EEs are also granted online and by mail. Online requests generally are done as disputes with either 'Too old to be reported' or 'Other' being the reason and a simple statement asking for an EE.