It depends on the era and the composer in question. Of the composers you named, most are from the 18th century (Mozart, Beethoven, Haydn), so let's start there.

In the eighteenth century, exposure to non-European music would have been limited and, perhaps more importantly, highly filtered. There is an account of French composer Jean-Philippe Rameau hearing an exhibition of song and dance performed by two Native Americans in France in 1725. (More on that and French responses to Native American music here). Rameau then reflected on that experience in writing some of the music to his opera Les indes galantes. But the music is very, very different from what he would have heard.

Europeans had some opportunities to hear Turkish military bands during the 17th and 18th century, both during war with the Ottoman Empire and (probably more pertinently) during diplomatic events. These sounds, again filtered through European imaginations, became the basis for the 'alla turca' style that you would recognize in Mozart's Rondo alla Turca. It's also the sound in the first variation of Beethoven's 9th Symphony, Movement IV. 

Beyond these instances, European composers were exposed to non-European music through accounts from travelers going to other places and coming back to Europe. There's an interesting exhibit about European impressions of Chinese music in the 18th century, which were based largely on accounts from Jesuit missionaries. But in order to transmit the music, those missionaries had to write it down in European notation, which distorted the music. Some of those pieces were then arranged for home performance, further distorting the sound. 

During the 19th century, as transportation technologies got faster and as European imperialism grew, European composers had more and more opportunities to hear non-European musicians performing in Europe. The World's Fairs presented a lot of these opportunities. Debussy famously heard Indonesian gamelan during the 1889 Paris World's Fair and that had an impact on the way he wrote music, especially his use of pentatonic and whole tone scales. During this time, many European composers continued to read accounts of travelers describing other musics, again always filtered through European notation until the development of recording technology. There's an interesting new book on Russian musical ethnographies and the impact those ethnographies had on Russian composers like Balakirev and Rimsky-Korsakov.

It's also a mistake to imagine that this transmission was only one sided. Indonesian gamelan, for example, was influenced by European music. There's an excellent new open source textbook on this phenomenon of musical transmission that I highly recommend by Danielle Fosler-Lussier, a professor at the Ohio State University. This book also covers Liszt, who you mentioned, and his relationship to Romani music. That music is European— Romani people lived in Europe—but it's still different from European classical music.