"becomes elusive" is a perfectly valid and usable part of a sentence in written English.
You can use it to describe something or someone becoming more difficult to find. For example, "The truth becomes elusive the more one delves into the ordeal."
Data is also becoming ubiquitous just as privacy becomes elusive.
And the promise of American citizenship or residency becomes elusive.
It also changes the larger world of the ballet, and perhaps helps explain why this woman (and on some mysterious level all the women) later becomes elusive.
"Where do we go when motive becomes elusive?" In a bar scene, Holden and his new mentor swiftly restate the theme over bottled beer.
"The R.S.C. has struck gold," declared John Gross in The Sunday Telegraph, adding that even when "The Family Reunion" becomes elusive or off-putting, it is "curiously compelling -- you succumb to its spell".
All this leaves Speaker John A. Boehner with essentially two choices: pass bills with House Democrats, which is the political equivalent of cheering with Phillies fans at a Nationals game, or let his conference pass bills that are so far to the right of anything that the Senate passes that compromise via conference committee becomes elusive.
Deep breaths become elusive.
But after he was discharged, in December, he became elusive.
For some, though, any semblance of security has become elusive.
Carmakers and other manufacturers will no longer extend credit (see article) and loans will become elusive and expensive.
Marketing executives who respond positively to my initial inquiry become elusive when the c-word is mentioned.
