Mostly tenders as far as I know.
Some of the operators work for mutiple 'rival' integrators; Swiftair have a FedEx feeder contract along with work for TNT and DHL.
On the other hand, Star Air narrowly re-won the UPS Cologne-based 10-year tender against ASL and West Air, and whilst they also do work for non-integrator Lufthansa Cargo the rumours are that UPS are very strict about their operations and proprietary processes and so forbid tendering for other similar work.
In Western Europe, Eurotrans and TNT Airways are sole-customer orientated but it's not reciprocated by the integrators; for TNT I believe the various affiliates ( Bluebird, West Air, Jet Time, Cygnus etc ) can bid for work. I don't know the exact mechanisms of it.
I'm sure an expert will be along shortly to correct everything I've said!