Do Ainur have wings?

[Estelyn Telcontar]

"Of course not!" I can hear indignant voices and thoughts saying. And of course I know Tolkien's works well enough to realize that there is no mention of winged Ainur in any of them. And though the thread title is a tongue-in-cheek parody on the Balrog wings discussion, the topic is a serious one. Please bear with me while I explain my thoughts.

The Ainur are similar to Judaic/Christian angels, with some elements of mythological heathen gods thrown in. Angels are generally perceived as winged, though there is little (perhaps even no? I'd have to look it up to be sure...) actual mention of that in the Bible. The reason for that notion is their ability to travel quickly, going back and forth from heaven to earth, or location to location, in almost no time. As to the mythological gods, there are some who are winged, most notably Mercury/Hermes. I assume that fast, airborne transportation would be the reason for that as well.

Both Christian and mythological elements being the background for Tolkien's Ainur, why do none of them have wings? And which method of transportation do they then use? A brief skim through "The Istari" in UT did not give me any information about how they traveled; it only says that they came to Middle-earth.

I assume that Ulmo, Ossë and Uinen swam, since the waters were their domain. Tulkas "can outrun all things that go on foot", and Oromë rides a horse. Other than that, no other methods of transportation occur to me at the moment. (I do hope that those more knowledgeable in the Legendarium will add more references if they know of any.)

Now, if they originally travelled throughout Middle-earth, how? Perhaps their non-meddling policy in the latter times was due to the fact that they could not easily get 'there and back again'? Or did they, as spirit-beings, take on their forms after arrival?

Both source information and speculative thoughts are welcome here!