One of the most common questions people have about Holistic Management is, "Will it increase methane?" Methane is a potent greenhouse gas and cattle produce a lot of it. Some grazing animals produce less than cattle — camels, for example, although it's because their metabolism is lower, so they actually produce as much methane per pound of feed. They just eat less (source).
animals are grazed using Holistic Management, however, methane may not
be an issue, for various reasons, including the fact that fresh grasses
are digested better than feedlot grain, and that the manure is in the
open on the range where bacteria can break down the methane rather than
put into a "manure lagoon" as is often the case with feedlot cattle.
if animals get more Omega 3 fatty acids from their food, they produce
less methane, and a healthy pasture like that produced using Holistic
Management includes grasses with higher Omega 3 content than the grain
mixtures typically used in feedlots (source).
And, of course, scientists are looking to breed cattle to produce less methane (source).
Read the Holistic Management's white paper on methane here (PDF): An Exploration of Methane and Properly Managed Livestock through Holistic Management.