The resources in a multiverse are merely uncountably great, and therefore civilisations that differ in their values and goals may well come into conflict over this limited pool, futile as becoming the ruler of a single thread in the timeskein seems from this far out.
Of course, they take themselves seriously, which is reason enough to take them seriously. Let us begin.
Early Nudge War
In the early nudge era, civilisations don't have the capability to shatter universes, and rely exclusively on sub-universal weapons. Warfare in this case is exclusively between adjecent universes - distant universes require exponentially more energy to communicate with, which means that supply chains are uneconomically stretched.
War is waged through two methods. Either nudges are used to directly disrupt enemy infrastructure, or used to assemble replicators in enemy universes which then destroy enemy infrastructure. The former is comparatively more effective against well-developed universes, and the latter more effective against sparsely populated universes.
Defence in this case involves making sufficiently robust intrauniversal infrastructure, with self-repair capabilities or an immune system to deal with hostile.
Late Nudge War
By the late nudge era, civilations have better control over the multiversal fields emitted from the membranes of universes, including an understanding of their refractive properties. Warfare is still between adjacent universes, though can take place over larger scales.
Typically, at this point, directly attacking universes with nudges dominates over assembling structures in enemy universes.