Here are some strategies for some of the Tournament levels:

DM-Stalwart - In the garage with the Pulse Gun (NOT the Flak Cannon), a solitary crate by the windows can be opened. A Big Keg o' Health is inside.

DM-Fractal - In the centre of the arena, on the trapdoor is a Shield Belt. Always camp near this and don't let Luthienne anywhere near it.

DOM-Condemned - On the rooftop, to the left of the Garage Point is a broken window. Inside is another Big Keg o' Health.

CTF-Niven - Under the large ramps in the area with the big gears, you can find both kinds of armour.

CTF-Face (Facing Worlds) - Search the teleporters for two Sniper Rifles and a Redeemer.

AS-Frigate - Get your bots to attack the base, then dive into the water. Swim under the Frigate and in via the lower room with the Flak Cannon.