I can't speak to the rest of the world but in the US it was a fairly logical transition, managed by the federal government over many decades as Congress expanded the role of government in maintaining public lands and landmarks. A fair number of historic battlefields/military history sites were managed/owned/interpreted by the US War Department early in the 19th century. When the first National Park was established in 1872-- Yellowstone --there was no agency to manage it directly; the Secretary of the Interior was given authority to determine how it would be managed. Congress appropriated very little money for the early parks, so the US Army actually ended up managing Yellowstone from 1886 until the National Park Service was established thirty years later. During this period the Army's primary duty was to protect the park's resources from exploitation (poaching, timber theft, squatting, etc.). Karl Jacoby's Crimes against Nature: Squatters, Poachers, Thieves, and the Hidden History of American Conservation has a great chapter on this period. Mission #1: nature & wilderness preservation. (National Parks)

In 1906 Congress expanded the government's preservation program through a piece of legislation called the Antiquities Act. Originally promoted by archaeologists (a newly organized field of professionals), the Act was intended primarily to protect Native American sites from looters-- the pot hunting trade in particular was booming, and looters were destroying sites around the Southwest digging for pots to sell to collectors. The Antiquities Act clearly shows Congress's intent was to protect archaeological sites by establishing a new category of reserves to be called National Monuments, "the limits of which in all cases shall be confined to the smallest area compatible with the proper care and management of the objects to the protected." So they were to be small, like the cliff dwellings of AZ/NM. But the law didn't just cover archaeological resources, it mentioned "historic landmarks, historic and prehistoric structures, and other objects of historic or scientific interest." The "scientific interest" qualification was very broad, so Theodore Roosevelt used the law widely, naming Devil's Tower (in Wyoming) the first National Monument based on its geological interest; he later did the same with the Grand Canyon, which was far larger than Congress had imagined. See Hal Rothman's classic study America's National Monuments: The Politics of Preservation for details. Mission #2: archaeolgical and scientific preservation. (National Monuments)

The National Park Service was established by Congress to manage both the National Parks and National Monuments in 1916. The following year, internal debates over the mission and function of the new agency was settled by the NPS's first Director, Stephen Mather, in the form of a letter from Interior Secretary Franklin Lane (later of Teapot Dome fame) outlining his responsibilities and the operating philosophy of the new park system. Interestingly, however, the so-called "Lane Letter" was actually drafted by Mather himself, effectively writing his own ideas about park priorities into policy. Read it and you will see the mission of the NPS at that point was primarily about wilderness preservation, not historical sites or battlefields. Al Runte's National Parks: The American Experience provides a fine explanation of this period and Mather's influence on the formation of the NPS.

Finally, to get to OP's question most directly, we have to skip ahead to the Great Depression. Stephen Mather was dead, and his lieutenant Horace Albright was running the agency. Congress gave FDR wide authority in reorganizing the federal government in 1933, and Albright convinced the President to use that authority to consolidate all of the country's preservation work under a single agency: the National Park Service. This was done through what is called the "Reorganization Act of 1933," Executive Order 6166, and further detailed in Executive Order 6228, in the summer of 1933. These orders effectively ended the War Department's management of historical battlefields, historic sites, and cemetaries, transferring their management to the National Park Service. Mission #3: commemoration and maintenance of national memorials. (National Memorials)

The list of sites transferred from the War Department to the NPS (in the Interior Department) following executive order No. 6166 is impressive. It included the following:

NATIONAL MILITARY PARKS

• Chickamauga and Chattanooga National Military Park, Georgia and Tennessee.

• Fort Donelson National Military Park, Tennessee.

• Fredericksburg and Spotsylvania County Battle Fields Memorial, Virginia.

• Gettysburg National Military Park, Pennsylvania.

• Guilford Courthouse National Military Park, North Carolina.

• Kings Mountain National Military Park, South Carolina.

• Moores Creek National Military Park, North Carolina.

• Petersburg National Military Park, Virginia.

• Shiloh National Military Park, Tennessee.

• Stones River National Military Park, Tennessee.

• Vicksburg National Military Park, Mississippi.

NATIONAL PARKS

• Abraham Lincoln National Park, Kentucky.

• Fort McHenry National Park, Maryland.

BATTLEFIELD SITES

• Antietam Battlefield, Maryland.

• Appomattox, Virginia.

• Brices Cross Roads, Mississippi.

• Chalmette Monument and Grounds, Louisiana.

• Cowpens, South Carolina.

• Fort Necessity, Wharton County, Pennsylvania.

• Kenesaw Mountain, Georgia.

• Monocacy, Maryland.

• Tupelo, Mississippi.

• White Plains, New York.

NATIONAL MONUMENTS

• Big Hole Battlefield, Beaverhead County, Montana.

• Cabrillo Monument, Ft. Rosecrans, California.

• Castle Pinckney, Charleston, South Carolina.

• Father Millet Cross, Fort Niagara, New York.

• Fort Marion, St. Augustine, Florida.

• Fort Matanzas, Florida.

• Fort Pulaski, Georgia.

• Meriwether Lewis, Hardin County, Tennessee.

• Mound City Group, Chillicothe, Ohio.

• Statue of Liberty, Fort Wood, New York.

MISCELLANEOUS MEMORIALS

• Camp Blount Tablets, Lincoln County, Tennessee.

• Kill Devil Hill Monument, Kitty Hawk, North Carolina.

• New Echota Marker, Georgia.

• Lee Mansion, Arlington National Cemetery, Virginia.

• Battleground, District of Columbia.

• Antietam, (Sharpsburg) Maryland.

• Vicksburg, Mississippi.

• Gettysburg, Pennsylvania.

• Chattanooga, Tennessee.

• Fort Donelson, (Dover) Tennessee.

• Shiloh, (Pittsburg Landing) Tennessee.

• Stones River, (Murfreesboro) Tennessee.

• Fredericksburg, Virginia.

• Poplar Grove, (Petersburg) Virginia.

• Yorktown, Virginia.

From 1933 onward, American military memorials, battlefields, and monuments have been largely managed by the National Park Service according to that agency's (ever changing) interpretive guidelines. As such, they share similar approaches, didactic techniques, design elements, and educational goals. The agency eventually established separate, detailed internal guidelines for managing the three majors types of resources (natural, scientific/archaeological, and memorial); to these a fourth category was added in 1964-- recreation --following a massive national study by the Outdoor Recreation Resources Review Commission, chaired by Laurance Rockefeller, which found that many Americans lacked recreational opportunities and argued for federal action to expand these closer to urban population centers. This led to the creation of new kinds of NPS assets like Golden Gate National Recreation Area and Gateway National Recreation Area, neither of which were wilderness, archaeological, or memorial in nature; they were simply created to provide recreational opportunities to nearby urban populations. Earlier coastal reserves, such as Padre Island National Seashore, Cape Cod National Seashore, and Indiana Dunes National Lakeshore stemmed from the same impulse. Mission #4: recreation. (National Recreation Areas, National Seashores)

So to get back to OP's initial question, these resources were either owned by the federal government initially or purchased from private parties for the purpose of preservation. Many were originally managed by the War Department (which did little in the way of interpretation) but after 1933 almost all in the US were managed by the National Park Service, which had clear policies and practices related to both preseration and interpretation that helped give them all a somewhat similar "feel." This is particularly true after 1956, when Congress appropriated $1 billion for a park expansion/upgrade program called "Mission 66" that resulted in massive contstruction, improvements, and new interpretive efforts throughout the system. Even today it is easy to pick out the Mission 66 visitors' centers or at times recognize the very stale interpretative strategies developed with those funds in the 1960s (and never updated).