Participles only express time, i.e., can be said to have tense, relative to the finite verb in the main clause; they have no effect on narrative time.
A present participle indicates a concurrent action or state. A person who sat at a table doing this or that is sitting and doing at the same time:
There was a stack of new lumber against one wall and a girl was sitting on it wearing a red hat. A baseball cap-type hat. — Terry Bisson, In the Upper Room and Other Likely Stories, 2000.
Wearing a black hat, black veil, black dress, black gloves, and supported by her houseman Brownsmith, she made a tragic figure. — Janet Dailey, Green Calder Grass, 2003.
A past participle of a transitive verb, such as supported in this sentence, is always passive, here including the agent, and is primarily descriptive, expressing no particular relationship to the finite verb. Past participles of intransitive verbs merely describe a state.
Especially in an absolute construction, however, a past participle may signal prior time, but no more than the usual narrative sequence of “That happened, then this happened”:
Their task completed, the grounded geese launch out on their own or join another flock. — Fern Reed Yarnick, His Touch, 2004.
His money gone, Louie tapped a friend for a thousand-dollar loan, staking his car as collateral. — Laura Hillenbrand, Unbroken, 2014.
If prior time is topical, then the only solution is a perfect passive participle where the form of to be is explicit:
Three beautiful mummy-cases were laid open, an ordinary mummy was placed in the last, the original one having been previously pillaged… —Thomas Joseph Pettigrew, A History of Egyptian Mummies, 1834 (dig. repro., 2013).
As you can tell by the original publication date, the perfect passive participle was more frequent in the 19th century than today, but writing “having been pillaged before” would make it seem somewhat less antique.
With transitive verbs, the perfect participle still finds use to express prior time relative to the finite verb:
Having finished their meal, they talked loudly, partly in Spanish, and then occasionally reverted back to their own native language. — Melvin Douglas Smart, The Hills of Death: Los Cerros de la Muerte, 2014.