"become narrower" is a correct and usable phrase in written English.
An example sentence would be: "The river's banks became narrower as we traveled upstream."
Sometimes, when our tastes become more defined, they become narrower.
As a pedestrian I dislike that pavements have become narrower to give space for car parking.
"The approach of the Republican party and Mr McCain has become narrower and narrower.
Sadly, the parameters of intellectual discourse in the West have become narrower and narrower.
If anything, they have become narrower, whiter, more male and more elite.
In a World Cup you can never underestimate your opponent because margins in football have become narrower and narrower.
In recent weeks, Mr. Powell added, "the approach of the Republican Party and Mr. McCain has become narrower and narrower".
Margins in improvements in performances have become narrower and narrower as athletes approach the limit of human achievement.
As the incidence angle becomes larger, the channel width become narrower, however the light tunneling channel number is unchanged.
The base of the leaves have turned from pink to green, for example, and the leaves have become narrower.
As a brisket tapers toward the back, slices become narrower and more interlaced with fat and connective tissue.
