An NYPD detective who chased down a Manhattan purse thief while on horseback was promoted Friday, as he humbly said the western flick-like pursuit was simply part of his job “to take action.”  Kyle McLaughlin, 36, was bumped up from detective to detective specialist following the April 15 arrest of Felicia Field – who is also a convicted killer on parole – at West 72nd Street and Columbus Avenue. “It’s awesome. It feels very great being promoted,” said McLaughlin, a married dad of four. “[This is] the greatest department in the world.” McLaughlin was riding his 23-year-old standardbred horse Kelly when bystanders flagged him down to alert him about a woman’s purse being stolen.  Det. Kyle McLaughlin, 36, joined by equine partner Kelly, was bumped up to detective specialist following the April 15 arrest of a purse snatcher at West 72nd Street and Columbus Avenue.…