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The phrase "in lockstep" is correct and usable in written English.
You can use it to describe two or more things that are moving in the same way or at the same speed. For example, "The French and German economies have been growing in lockstep since last year."
Exact(59)
"They're in lockstep," Mr. Collins said.
Global financial assets were moving in lockstep.
He votes in lockstep with the party.
Total operating expenses move up nearly in lockstep with sales.
POWER consumption used to march in lockstep with economic growth.
Not everyone was in lockstep with the party's new leader.
Art never shifts in lockstep from style to style.
The results, so far, are completely in lockstep.
"I don't think we will walk in lockstep.
Not all council members are in lockstep with this scheme.
But he did not move in lockstep with Republicans.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com