The phrase "become over time" is correct and usable in written English.
You can use it to describe how something changes or develops slowly over a period of time. For example: "Their relationship had become stronger over time."
Their priorities are not as common sense as they will become over time.
But then again, as we say in Pearl Jam Twenty, they have become over time the most dependably unpredictable band in rock.
Meditations on the frieze lead Harbison to Peter Weiss's "immense historico-political novel 'The Aesthetics of Resistance,' " a book composed of "unwieldy blocks" of prose, not unlike the unwieldy fragments of stone that the Pergamon frieze has become over time.
And the more the animals rely on display to elicit help or effort from a partner, said Dr. Wachtmeister, the more extreme and conspicuous those displays are likely to become over time.
–Azar Nafisi (Iran), author Lolita by Vladimir Nabokov (1955) The landscapes of honest colours, fata morganas and melting mirages have become over time an America more real than America itself, a reimagined America which in the end is, must be, the most enduring (and beautiful) country of all.
I'm especially skeptical of the instant-on stuff, given how bloated Windows installations tend to become over time.
This case became, over time, a kind of rotating media Ferris wheel.
The title "Senator" becomes, over time, less an honorific than an epithet.
They discovered that names which sounded more distinctively "black" became, over time, ever more reliable signals of socioeconomic status.
But he also became, over time, a kind of secret fetish among composers of a more progressive type.
