"engage throughout" is a correct and usable phrase in written English.
It means to actively participate or be involved in something for the entire duration. Example: "The students were encouraged to engage throughout the entire class discussion, sharing their thoughts and asking questions."
The two main characters in "Thinks" -- Ralph Messenger, a notorious womanizer and the director of a cognitive sciences center; and Helen Reed, a recently widowed novelist -- engage, throughout the novel, in flirtatious banter about their differing views of the world.
While we argue that formal education should be more like the informal learning processes in which people and other organisms engage throughout our lives, we are not advocating the abandonment of formal education and will turn to its distinctive features below.
In Rubio's view, insufficient belligerence by President Obama is part of a "historic reluctance to engage throughout the Middle East".
Ms. Tuff kept him engaged throughout the ordeal.
His charismatic manner kept me engaged throughout the entire seminar".
Parents are engaged throughout and must sign a consent form".
Supporting roles count for less than usual here, but the company is engaging throughout.
"Whatever It Takes" is engaged throughout, nowhere more so than in a vivid section on Baby College.
Ledger is plainly the star attraction – single-handedly keeping this viewer engaged throughout swaths of tower-top bat-brooding.
Mr. Moses' play is likable too, although it does not always deliver enough drama or insight to keep us fully engaged throughout its 95-minute running time.
England's left-wing was fully engaged throughout, leaping for the ball, tackling into touch, but he may hope that the citing officer does not dwell on a high, late, armless challenge on Paterson in the closing stages.
