The absolute construction usually means the equivalent of "while" (at the same time) or "because" (the main clause is an effect of or reaction to the absolute clause). Often both meanings are hazily suggested or implied.
Your first sentence, without "being", more strongly suggests the "while" meaning. It's roughly equivalent to this:
As the horsemen raced down the hill toward the settlers, their guns were at the ready.
Your second sentence, with "being", more strongly suggests the "because" meaning. It's roughly equivalent to this:
Now that their guns were at the ready, the horsemen raced down the hill toward the settlers.
As for why the weight between the meanings shifts in this way, I've never thought about this or even noticed it before you asked the question, but my first thought is that in your examples, "being" draws more attention to the absolute clause's meaning at the start of the action described by the main clause, and omitting "being" draws more attention to the absolute clause's meaning for the full duration of the main action. This is true even in your second two examples, where the absolute clause describes something being "over" before the main clause's action even starts. I doubt that it's a general rule, though.
It might help to look at old, influential precedents that contemporary usage echoes. The first well-known example that comes to my mind is the most infamous absolute construction in U.S. history, in the 2nd Amendment of the Constitution:
A well-regulated militia being necessary to the security of a free state, the right of the people to keep and bear arms shall not be infringed.
The word "being" here clearly gives the absolute construction the "because" meaning. This has fueled considerable debate about how to interpret the law: some say that the absolute construction implies that the 2nd Amendment extends only to protecting militias, some say it doesn't matter, some observe that in fact militias are no longer relevant to national defense, etc.
As for its relation to time, this absolute construction is meant as an eternal truth (even though militias in fact became obsolete), which holds before any infringement happens, as well as during. The emphasis is on the natural state of affairs that exists before an infringement.
There are probably better well-known examples to look at, but that one does nicely illustrate the role of "being". However, if "being" were removed, the sentence would become ungrammatical rather than changing meaning.
