With the expected breakdown of law and order, the economy and central governments the rise of bandits, raiders, low-level warlords and roving bands of warriors (not soldiers) will begin. This return of low and medium intensity warfare into the mainly civilian areas of countries has a few responses. Knights and a feudal system were highly successful, so were their counterparts in Japan, but for a primarily urban population, based in towns or cities there is another good option. Militia that only have to defend the city and the surrounding countryside, have the backing of the concentrated manpower, technical skills and resources of an urban area will be adequate to deal with low-medium intensity threats without needing the level of time and resource inputs of a professional army.
The primary mode of warfare, Sea and air are ultimately in the service of objectives on land in all but a handful of cases and provide logistical, reconnaissance or protection roles. Land warfare will be changed by peak oil but the question is how? The major components of modern land warfare are the gun, artillery (which is the same basic tech just bigger and slightly altered) and motorised vehicles. The gun is a relatively simple tech that’s been in use for at least 600 years in one form or another and a lot of the advancements aren’t dependent on fossil fuels but precision machining and metallurgy which can be preserved in a post peak world. Artillery, “queen of the battlefield” that is gunpowder based has been around just as long and is certainly possible on a low-tech scale (cannons have been made out of ice and wood). While the tech is slightly different from guns, especially in the modern forms, the principals are the same and the tech base as well. Motorized vehicles, either; transport, tanks or APCs, are different matter since there underlying tech is the combustion engine which is almost entirely fossil fuel dependent. The engines can be run on biofuels and most likely the future analogue of them will be powered by such but that will come with tradeoffs, especially in the world of logistics and the civilian support structure. These components are the ones most likely to suffer, both motorized vehicles and the support structures, and modern armies’ high consumption rate will be a hindrance to its ability to function in a post-peak world. Various solutions such as using bikes, increased train use and reduced consumption of key resources could help mitigate these problems.
Naval warfare is very different from land and has its own problems and strengths. First, navies are expensive and you get what you pay for, cheap navies are highly ineffective at even pirate hunting and are barely worth the cost. Good navies can be built at a lower resource and tech level but the infrastructure necessary doesn’t exist as now. So what will post-peak navies look like? Two things define naval eras; their weapons (boarding, ram, cannons, turrets, etc) and their main source of motive power (oar, sails or engine). Oars for warships only make sense in close waters, like the Mediterranean (there were oar-powered galleys in the Crimea war). Wind has the advantage of being free, the sails were expensive though, and were still used on the first steam powered warships to give strategic advantages (lower fuel use) so it should steadily become more important. While engines have tactical advantages and aren’t dependent on the vagaries of the wind they will lose prominence due to the increasing need to use biofuels which are expensive, limited and less energy dense. My bet is that engines will stay on coastal ships, think torpedo boats or some classes of destroyers, and potentially the big ships of the line that will come back with the changes in aircraft and missile tech. Weapon systems will most likely shift back to shells delivered by cannons or turret and away from missiles as the needed tech fades away. Metal, or metal clad, hulls will be essential since explosive shells, which are not a fossil fuel dependent technology, because shells shred wooden ships (maybe construction tricks could change that but I apart from cross laminated timber I haven’t heard anything that could potentially do that).
The only unique form of warfare the industrial age has produced (space warfare hasn’t happened yet) has several pathways available to it. The simplest is that the tech and abilities stop existing completely and in some places this is a certainty especially where its use in warfare is minimal and no domestic use is available. It can also stagnate in its current form; this is unlikely due to the jet engines reliance on high fossil fuel consumption that biofuels won’t be able to substitute. It can regress into a much earlier form, this will certainly happen in some parts of the world and even advancing down other paths will look like this in some way. Lighter than air could quite easily make a comeback in this situation, especially with the decline of missiles, and its advantages will count for quite a lot. The other is alternate forms of air travel that either haven’t been invented yet or have only been tried now, the synergy aircraft project or solar planes for example, and would allow a dynamic airforce. Buts its role and abilities will be diminished and its current status as a separate force will almost certainly disappear and it will become an auxiliary of armies, similar to artillery and other attachments, and for navies, escort carriers or island/shore based. Its role will also change back to mainly reconnaissance with relatively minor transport, attack or defence roles (there will still be dogfights, just in a different setting).