TL;DR- The revolver was the preferred service pistol by LEO’s from its first use in 1847 up until the 1980’s. But in 1986 there was a shootout in Miami between two robbery suspects and the FBI and the FBI was outgunned. This led to the FBI wanting to do away with the revolver and to find something better. The FBI first adopted the 10mm S&W Model 1076, but then switched to the SIG Sauer P226 and then the P228. In 1982, Glock released the Glock 17 and at some point in the mid 80’s, due to its superiority and advantages when compared to other handguns was adopted by the St. Paul and Miami police departments. In 1994 the NYPD adopted it too and began issuing it. Today, 65% of all LEO’s use a Glock. Overall, due to how cumbersome loading a revolver was during a fight and other tactical advantages offered by semi-automatic pistols, LEO’s made the switch to them.

Originally, the first service pistol used by LEO was in 1789. The members of the US Marshals Service appointed by President George Washington carried single shot flintlocks. In 1836 the revolver was invented, which was far more practical than the single shot pistol. The first service revolver implemented was used by NCO’s in the military. The first LEO use of the revolver was by the Texas Rangers shortly after the military’s adoption of it. Under Captain John Coffee “Jack” Hays, the Rangers adopted the Colt Paterson five-shot .36 caliber single-action revolver. The Rangers, in 1847, phased out the use of the five-shot in favor of the six-shot .44 caliber Walker Colt single-action revolver. In 1835, Boston formed the first police department and in 1845 New York established one too. And in 1885, the New York police commissioner at the time, Theodore Roosevelt, began to issue Colt New Police .32 caliber, six-round double-action revolvers with 4-inch barrels. And in 1934, one year after the repeal of prohibition, the FBI officially armed its graduating class of agents with Colt Official Police .38 Specials. Up until the 1980’s, the magnum revolver had become the preferred LEO’s service pistol. Although, the first real use of semi-automatic pistols was by the Illinois State Patrol, which armed it’s officers with the S&W Model 39 for a decade starting in 1967. During the time between 1965 and the 80’s some officers did use semi-automatic pistols but were very much overshadowed by the use of the revolver. It wasn’t until the 1980’s were LEO’s were being outgunned that they made the switch to semi-automatic pistols. The example I gave above was the 1986 Miami shootout. But the real game changer was the Austrian-made Glock. Offering up to 17 rounds, a mere 5-pound trigger pull, lighter body weight due to polymers and fully interchangeable parts, the Glock promised firepower, accurate ease of use and affordability.

Just a side note: revolvers are types of pistols. I think you mean "from revolvers to semi-automatic" (semi-automatic meaning the next round is automatically loaded in to the chamber by the force of the recoil).