Brewer's: Walk through One's Part
(To). A theatrical phrase, meaning to repeat one's part at rehearsal verbally, but without dressing for it or acting it. To do anything appointed you in a listless indifferent manner.
“A fit of dulness, such as will at times creep over all the professors of the fine arts, arising either from $$$ or contempt of the present audience, or that caprice which tempts painters, musicians, and great actors ... to walk through their parts, instead of exerting themselves with the energy which acquired their fame.” —Sir W. Scott: Redgauntlet, chap. xix.