All modern monuments should be installed on a solid concrete foundation. The foundation should be dug as deep as the frost line, depending on the locale. This would be about the same depth footings for residential construction is dug. In the Northeast this means footings should be at least 42 inches or more.
The funny thing about a foundation is that once it is poured, you can only see the top of it. Once the monument has been installed, you can not see any part of it. Only the person who dug it knows how deep it really is, and only they know what the hole was filled with. The largest cause of leaning and fallen monuments may be attributed to poor quality foundation work.
Older monuments are likely to have a lose set rock footing, or none at all. They are at the mercy of the grade they are installed at, and the soil conditions themselves. When resetting historic monuments it may be advisable to install a new concrete foundation, even if it never had one. This will greatly increase the cost of the reset, but will also make it more permanent.
Congressional Cemetery, in Washington DC, has lose set, brick foundations under nearly all of its monuments. Some of these have lasted better then others, but on level ground they are mostly all still level . An advantage of a lose brick footing is, if the footing ever heaves, (raises from frost) once the monuments has been moved, bricks may be removed to re-level. If badly heaved, a solid concrete footing would have to be jack-hammered down, or entirely removed to be lowered.
If a new footing is not installed, a bed of crushed stone, or sand mixed with pea stone gravel, should be placed and packed into a level position. Then the monuments base may be reset followed by the successive monumental sections, until the resetting operation is complete.