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It was the first time a player in international cricket was substituted under the new concussion rules.
Initially, Smith passed a concussion test and was allowed to return to the field and complete his innings, but his condition worsened the following day.
Brain injury charity Headway said it was "incredibly dangerous" for Smith to have resumed his innings.
"You need to take an 'if in doubt' approach," Luke Griggs, Headway deputy chief executive, said.
"With concussion, the vision can be blurred and the brain can be slow at processing information. That leads to delayed reaction times and is just incredibly dangerous."
Smith scored 142, 144 and 92 in his three innings in the Ashes series - more than a third of Australia's runs in those innings - and is ranked second in the ICC test batting rankings.
He has scored 243 runs more than any other visiting batsman so far in the series.
Apart from Smith, only one Australian batsman has scored a century (Matthew Wade) and Travis Head and Labuschagne have scored fifties.
Labuschagne is the most likely option to replace Smith for the Headingley test starting on Thursday, having made 59 as Australia battled on Sunday to a draw which retained their 1-0 lead in the series.
Meanwhile, England have named an unchanged 12-man squad for the third test as seamer James Anderson continues to recover from a calf injury.
Smith has two weeks to recover and prepare for the fourth test in Manchester, which begins on September 4.
Australia also play a three-day tour match in Derby from August 29-31.