I'm guessing recruitment given the phrase "...for their selection process..."
I can't speak to recruitment from a place of expertise, given the only outsourcing the airlines I've worked for were for drug testing. When outsourcing training though, I would supply the likes of CAE or FlightSafety with the training and checking program I wanted used. Some airlines accepted their core programs, ours didn't. For example, I'd tell them I wanted x-number of hours of ground school, or that these were the y-exercises I wanted in sim session c. We would then work back and forth putting the program together. Afterwards, I would use student feedback and results to get a feeling of how the program was running, and once in a while I'd go down to the facility to conduct an audit. Fairly straight forward, but there were certain centres that always gave me a headache because they wouldn't follow the program.
Putting my thinking cap on, I'd say that if I were to do the same for recruitment, I'd first put together a scripted program, making it easier to evaluate candidates based on their performance. This would include what exercises to run, what questions to ask, how long the test would take, and the grading parameters used. I'd then evaluate the number of successful candidates through the selection process to the number of successful pilots to complete the training program. I'd look at feedback from instructors, check pilots, and line managers - are there trends in the type of issues they are seeing, an increase in the number of unwanted personalities making it through, or is there an increase in repeated exercises or flight tests? Finally, I'd schedule a number of scheduled audits to talk with the company and obtain their comments, as well as a number of unscheduled audits to test the system.