The truth behind the story of "The Muncaster Boggle" is unlikely ever to be known in detail, for reasons which will become apparent; but here is all the information I have found, mostly from the Coroner's inquest.

Cumberland Pacquet, 19 Nov 1805 [though this was a day after the inquest, news of the proceedings had not reached Whitehaven by the time the paper went to press]:
"A young woman, of the name of Bragg, (a servant in Ravenglass) after being missing for several days, was found dead last Thursday [sic] by the side of the Mite [sic], at a place called Muncaster Steps.- A strong suspicion prevails that she had been strangled; but no discovery has yet been made of the perpetrator of this atrocious deed."

The coroner's inquest, 18 Nov 1805 (Whitehaven Record Office document D/Lec/CRI/114/9):

a) Witness statements [Mary's surname is accidentally given in the heading as Birkett]:

• Ann Bartram, wife of James Bartram of Ravenglass. Mary Bragg had been at the Bartrams' house on Friday "gone sennight" [i.e. 8 November], arriving about 5 or 6pm and staying until 9 or 10. Mary was wearing a "Dark Stamped Bedgown", a blue cloth petticoat and a blue brat apron, but no cap (she may or may not have had a neck-cloth). Mary "had no Work along with her", but as her master and mistress had been away on a visit since Wednesday, she helped Ann, who had some visitors (Ann's own servants were absent, as it was "the term time for changing Serv'ts"). Mary seemed as cheerful as usual, and when she returned to her employer Mr Grice's house [see below] next door, Ann saw her go into the yard and turn the lock of the yard door, but could not hear whether she opened the door into the house (although there were no other servants at home, and Mary had nervously arranged to share a bed with one of Ann's girls, she did so for the Wednesday night only, and expressed no concern on the Friday). As Mary was carrying a candle, Ann could see that she went into the porch, but the height of the yard wall made it impossible to see whether she actually entered the house.
  At noon the next day, Ann realised that she had not seen any sign that Mary was awake, and asked her family if they had noticed. As nobody had, Ann went to the Grices' to check. She found the door locked, and the window shutters unopened, but was unable to get any reply from within, so her husband climbed a ladder and looked into Mary's bedroom window, finding that the bedclothes had been turned down, which suggested that she had at least been to bed. A search was made, but she was not found until her body turned up in the Esk the following Friday [15 November]. Ann saw the body after it was brought to Ravenglass, and believes that the clothes were the same ones Mary had been wearing when she last saw her alive.
• James Bartram confirmed Mary's movements as far as he knew them. When Mary had failed to answer knocks at the door on the Saturday morning, he had checked and found that the porch door (which locked from the inside with a bolt) was unlocked, but the door from the porch into the kitchen was locked; following that he made his ladder inspection. Mr Grice came home later on the Saturday, and on entering the house, everything seemed in order; both he and Bartram were satisfied, on close examination of Mary's bed, that it had been slept in since it was last made. When he heard of the finding of Mary's body the following Friday, James took a horse and cart to bring her back to Ravenglass for the inquest; when found, she had a key to the yard door of Mr Grice's house fastened to the string of her brat apron with a draw knot. Although she was wearing a neck-cloth, it did not seem unusually tight.
• John Bragg & John Taylor, surgeons of Ravenglass, examined the body. They found no marks of violence that could explain her death, and confirmed that the blue linen handkerchief around her neck was, even when soaked, "so loose and fastened in such a manner that it could not occasion her Death or have been the means of strangling her". There were no marks on the neck to suggest that the handkerchief had at any time been pulled tighter to strangle her, and they would also have expected her face to be a different colour if she had been strangled rather than drowned. There were, however, some marks on her arms with "the appearance of being made or caused by the presence of fingers while the deceased was living" which the doctors could not explain, but seemed most likely to have been caused by Mary herself, for there were no corresponding marks of thumbs on the opposite side of the arms, as would have been expected if someone gripped her.
• Ann Thomlinson of Ravenglass deposed that about 4 or 5am on the morning of Saturday sennight [i.e. 9 November] she had heard some shrieking coming from out in the street, at the doors of James Barton's [sic] house, near the entrance to Mr Grice's yard, which she assumed was "some Lads and Lasses going to Fair" (the Martinmas hiring fair for servants, at Egremont) for it sounded much more like "young people in frolick" than cries of distress, so she ignored it.
• James Nicholson testified that about 4.30am on the morning of Saturay sennight he was working in Mr Frears' cow-house, when he heard two shrieks from the direction of James Barton's barn (probably not from the street). He thought it sounded like somebody frightened, but he did not recognise the voice, and could not see anybody, so he thought no more of it.
• Henry Bragg, Mary's father, "saith that he believes verily and truly that Sarah Jackson of Muncaster House and John Pickthall of the same place can give material information and are necessary witnesses"
• Mary Taylor of Ravenglass testified that on Friday last [i.e. 15 November] she, Eleanor Parker and Ann Armstrong were on their way from Muncaster House to work getting sea slack [?], when below Muncaster Steps they saw a hat lying on a shingle bank in the Esk channel (the tide then being fairly well out). Ann waded in and fetched it, finding that the upper part of it was dry, and that it did not seem to have spent much time in the water. Then about 100 to 150 yards further downstream, Eleanor saw something like a rope in the water, which they went to see more closely. It was then they saw a pair of feet pointing up-river, and the legs, and immediately decided to fetch Thomas Blackadder from Muncaster. He brought help, and Charles Shepherd and others went into the water and brought out the body, which all quickly realised was the girl who had gone missing a week previously. They saw the blue handkerchief on her neck "twice round loosely" and the sand in the folds of Mary's clothes. Mary witnessed the removal of the body to Ravenglass in James Bartram's cart. Mary could not recollect ever previously having seen the hat, or the neck-handkerchief, and could not say whether they had been Mary Bragg's property.