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Investigators are working to determine whether the man was brought into the U.S. from Canada illegally.
One incident has been reported in Bellingham and three similar incidents were reported in Bellevue.
Prosecutors said they will seek the death penalty if Kohberger is convicted.
The man was taken back into custody and booked on additional charges after fleeing from police while temporarily released from jail.
Anyone who sees suspicious activity in the construction area or anywhere else on campus is asked to call university police at 360-650-3555 or 9-1-1.
The suspect was arrested without incident and booked into the Skagit County Community Justice Center.
Investigators said three other officers had not violated policies and the fourth case was ruled inconclusive.
The attacker was last seen fleeing the area near I-5.
The man continued to use derogatory language towards African Americans when he was detained by Bellingham Police.
Bellingham Police say no arrests have been made and no charges have been filed yet.