From John Everett Millais 1878 painting.

The princes: who were they and where were they?
Edward (age 12), oldest son of Edward IV and Elizabeth Woodville was in Ludlow with Anthony Woodville, Earl of Rivers–Elizabeth Woodville’s brother. 24 April 1483, Edward left Ludlow in his uncle’s custody and headed for London where Edward is to be crowned King. By 1 May 1483, Richard, Duke of Gloucester and protector to the future king, takes custody of  Edward in Stony Stratford. They arrived in  London 5 May 1483, the day Elizabeth had tried to set for her son’s coronation. Richard sets it back to 26 June 1483 and installs Edward in the Royal Apartments in the Tower of London–the customary residence for kings awaiting coronation. Meanwhile, Elizabeth went into sanctuary at Westminster Abbey 1 May 1483, taking her other children with her, including her younger son, Richard.
Richard (age 10), younger son of Edward IV and Elizabeth Woodville lived in court. He went to sanctuary when his mother did and stayed there until 16 June 1483 when Richard, Duke of Gloucester and protector with the help of the Archbishop of Canterbury convinced her he’d be safe in the Royal apartments with his brother Edward.
On 26 June 1483, all of Elizabeth’s children were bastardized after Bishop Stillington’s revelation that Edward IV had a precontract with Eleanor Butler predating his marriage to Elizabeth Woodville. Then, precontracts were as binding as marriages. Neither boy could inherit the throne. They were moved to the Garden Tower as Richard III took up residence with his wife, Anne in the royal apartments awaiting his coronation.
The boys were reportedly seen playing on the Tower grounds throughout the summer, long after Richard had left on his progression through England. Some reported they had disappeared as early as June that year, while others claim to have seen them into February of the next year, but the consensus is they disappeared sometime in September, 1483. Richard III’s wardrobe account has an expense for Edward’s clothes to wear while attending Richard’s coronation 6 July 1483. Richard’s own son was north in Middleham during the coronation, so the clothes could only have been for the now bastardized prince. It is unknown whether Edward was actually in the coronation party.
In 1674, during a renovation of the tower, two skeletons were found near the White Tower when workmen removed a structure. The workmen discarded the bones in a rubbish pile and they remained there for several days before word reached King Charles II, who eventually declared that those were the remains of the princes. The bones now reside in an urn in Westminster Abbey and could be tested for a DNA match to Richard III. To date, the crown has not allowed for the bones to be removed for DNA analysis.
Whether the princes died under Richard III’s watch, or whether they survived him is not known. There is no extant documentary evidence that proves either scenario.