The moments leading up to the tragic death of Kimberly Burch -- fiancée of Faster Pussycat frontman Taime Downe -- have been captured on video, which is now in the hands of detectives ... TMZ has learned.

Sources with direct knowledge tell TMZ ... the Miami-Dade Sheriff's Office in Florida is in possession of video that shows Kimberly climbing the ship’s rail and taking a step off.

No official word yet on how Kimberly's death is being ruled. MDSO confirmed to TMZ Monday morning the investigation is still ongoing ... but from what it sounds like, the video speaks for itself.

MDSO's crime scene unit was at the Port of Miami Sunday morning when the Explorer of the Seas docked, to investigate the circumstances that led to Kimberly's death.

TMZ can confirm that Taime's bandmates disembarked the ship around 6:45 AM local time and waited for several hours for their lead singer to join them.

TMZ broke the story ... Kimberly went overboard last week following an argument with Taime ... with her mother confirming her passing. At the time, it was unclear whether Kimberly had fallen or jumped overboard.

Unfortunately, Kimberly's body has not yet been recovered.

Kimberly was accompanying Taime and Faster Pussycat on The 80s Cruise out of Miami.

Kimberly was only 56  ... RIP.