The Lauf "Grit" is the model of fork you will need for a gravel or cross bike. (ie. Salsa Warbird.) This is how I have my Warbird setup and LOVE it!!! As far as any "side-to-side" flex... if I stand up and mash hard and purposely sway my bike side-to-side, I can visually see some lateral flex. But I can't actually feel it. Keep your focus ahead where it should be and you will be happy. I even ride the Grit on road rides. Not necessary for such... but not worth the hassle to remove it.
The Salsa Cutthroat is a drop-bar mountain bike. The frame is designed to accept a mountain suspension fork. So the Lauf model you would need for this application is the Trail Racer. At 60mm, it has twice the travel as the Grit. And it is sprung slightly softer... because it is designed for trail use, not for gravel use. If I have anything against the TR, it is I wish it was sprung slightly stiffer. But that's on me. I'm using it for something it wasn't necessarily designed to do. That said, the Trail Racer shines on gravel. In fact, my Warbird hangs in the garage most days, because I usually opt for the Cutthroat / Trail Racer for most gravel rides. This is what I chose to take with me last weekend to Dirty Reiver in England. I got passed all day long on the climbs. (Because I'm fat and old.) But never once got passed on a descent. The geometry of the Cutthroat and the plushness of the Lauf Trail Racer made for an unbeatable combination on the long rough and choppy downhills.
DK has partnered with Lauf to form the "Lauf Dirty Dozen at Dirty Kanza". Twelve folks we hand-selected to share about their experiences with the Lauf fork as they prepare for, and participate in Dirty Kanza. You can read what they have to say here...
http://laufforks.com/Then click on "Just Ride" and scroll down the page.
Good luck.