There is nothing different about how a consultant makes money with Open Source Software (OSS) when compared to proprietary software, except that OSS allows the consultant to offer clients a much more robust set of software. Usually, this is because the consultant isn't limitted by license agreements with specific software manufacturers or distributors.
Also, since the OSS consultant has access to the source code, he can provide much better support. These consultants can look at exactly what a program is doing, rather than relying of someone else to tell him how it is supposed to work. This is one of the many reasons that we at Systuff suggest an Open Source Software solution whenever possible. It allows us to provide better service to our clients.