What are possible semantics of participle constructions?
As OP provides no actual examples, I've supplied actual sentences containing absolute clauses (usually of the form used in the famous 2nd Amendment).
First, there probably has to be a meaningful semantic connection between the absolute clause and the main clause, rather than mere synchrony:
?/* The stream gurgling softly, Eta Carinae exploded in a supernova. [semantic disconnect]
The 'plus' to synchrony may be quite subtle / mundane (but is probably usually more tangible than is often the case with participle clauses):
Riding my bike, I suddenly had the idea of starting a bridge club. [{participle clause}; mere concurrence (with same referent)]
at least when the absolute clause is placed first:
(perhaps?)The band playing 'Onward Christian Soldiers', the folk from Upperfield Methodist Church marched past.
The folk from Upperfield Methodist Church marched past, the band playing 'Onward Christian Soldiers'. (synchrony with semantic connection) .......
The rain obligingly holding off, England's batsmen are making steady progress. [concurrence but with an enabling (a necessary, but sadly not a sufficient, condition)]
Of course a sequentiality may be indicated:
Work being over for the day, most of the employees'cars were soon filtering out of the car park. (temporal; A being over, C came next)
The connection is perhaps a restatement/explanation/expansion upon the statement in the main clause (where fronted); a colon or dash (with minor rephrasing) could be used instead:
The weather was merciful on opening night, the rain obligingly holding off for the whole of the entrancing, and, at times, menacing walkabout performance. [David Nicholson_YorkMix]
Reason is another possible semantic connection:
Icebergs being common in polar oceans, many people are opting for cruises in these waters. [reason]
.........
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. [reason/'causative' (given certain governmental views)]
Cause is another possible semantic connection. Note here the idiosyncrasy of idiomaticity of variants using different time-frames:
The weather holding, forest fires are certain to break out. (ie As the weather is holding, forest fires are certain to break out.) [causative] [F prognostication]
With the weather holding, forest fires are certain to break out. [causative] [F prognostication, but the absolute is now connotatively less non-conditional]
.......
With the weather holding, forest fires are breaking out. [causative] [P]
??The weather holding, forest fires are breaking out. [causative] [P]
........
With the weather holding, forest fires were breaking out. [causative] [Past]
?The weather holding, forest fires were breaking out. [causative] [Past]
Conditionality is another possible semantic connection:
The weather holding, forest fires are certain to break out. (ie If the weather holds, forest fires are certain to break out.)
.......
The weather permitting, we will go to Southport tomorrow. [conditional/intent-choice]
.......
God willing, the weather holding, and the wind at our backs, we should be in ... port in just under two days. [The Rescue of Charles de Simpson: Book One in the Dorchester Chronicles By J.S. Witte][conditional]
Note that identical sentences can have different interpretations. Absolute constructions involve deletions, and it has often been said here that deletions often lead to ambiguities. And that context often disambiguates quite acceptably. And that care needs to be taken.
... apologies for no substantiating references on occasion