A global cat-and-mouse game involving the admitted leaker of National Security Agency secrets exploded into a diplomatic scramble, as U.S. authorities sought to catch Edward Snowden before he reached his next goal: political asylum in Ecuador.

Mr. Snowden's unexpected Sunday flight to Moscow from Hong Kong exposed the apparent limits of America's diplomatic and intelligence-gathering reach. At a time when Mr. Snowden has been the subject of intense interest from U.S. authorities, they were unable to prevent his departure from...