In Commander, do emblems from planeswalkers go to the command zone when a planeswalker leaves the battlefield?

Emblems are not permanents and never enter the battlefield at all. When a planeswalker's loyalty ability says 'you get an emblem,' the emblem is placed directly into the command zone as part of the ability's resolution (CR 113.1, CR 113.2). It does not pass through the battlefield on the way there.

Because the emblem starts in and remains in the command zone, the question of moving it when the planeswalker leaves the battlefield simply never arises. The planeswalker leaving has no effect on the emblem whatsoever (CR 406.7).

Emblems also cannot be moved, destroyed, exiled, or otherwise interacted with by most game effects. Once created, they persist for the rest of the game (CR 113.2).

Concrete example: You activate Elspeth, Sun's Champion's third loyalty ability (−7), creating an emblem that gives your creatures +2/+2 and flying. Your opponent then destroys Elspeth with a removal spell. The emblem remains in the command zone and continues granting the bonus to your creatures for the rest of the game — Elspeth's destruction has zero impact on the emblem.

In Commander specifically, CR 406.7 confirms the command zone is the correct and permanent home for emblems regardless of format, so no Commander-specific rule changes this interaction.