Among the examples of flying a kite was the suggestion in the 1960s by then Minister for Justice Brian Lenihan that the Republic of Ireland should rejoin the Commonwealth of Nations. He did so on the orders of then taoiseach Seán Lemass. However the reaction was negative and Lemass and Lenihan dropped the issue, with Lemass (with Lenihan's connivance) claiming that Lenihan had just been speaking theoretically in a personal manner and not for the government.
A famous British example occurred in 1885 when while his father's supposed approval, a son of then Leader of the Opposition William E. Gladstone revealed that Gladstone now supported home rule for Ireland. The revelation is known in history as the Hawarden Kite. Historians still debate whether the Hawarden Kite was a deliberate tactical revelation (a case of flying a kite) or an accidential revelation. Whatever the reason the reaction was sufficiently positive for Gladstone to publicly confirm the accuracy of the story and commit to a home rule policy.
One example happened in the country of Singapore in October 2007 - Minister Mentor Lee mentioned the idea of Singapore and the Federation of Malaysia remerging, but the idea was later met with some derision by both parties on either side of the Causeway.
An American equivalent is the term raising the flag and seeing who salutes (i.e., raise an issue and see the reaction).