ΔE 26.6, loose: different enough that you should pick on purpose. Battle Mud Wash is noticeably lighter than Agrax Earthshade.
✓ Two Thin Coats comparison chart lists these two as equivalents.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 27). Battle Mud Wash is noticeably lighter than Agrax Earthshade.
The same difference applies the other way round: Agrax Earthshade is noticeably darker.
ΔE is the colour difference between our two swatch estimates: under about 6 is close, over about 15 is loose. Real paints also differ in finish, coverage and how they thin, so test before you commit.
See Agrax Earthshade on the colour wheel See Battle Mud Wash on the colour wheel
All matches: Agrax Earthshade equivalents · Battle Mud Wash equivalents