pull, tug, jerk, admit, install, quote, refer to
曳 centers on pulling or dragging something with effort: a tug or jerk. The meanings 'admit', 'install', 'quote', and 'refer to' are less common and may represent extended uses where something is drawn in or brought forward.
towing (a ship)
festival float
something so heavy it must be pulled by a thousand men; pulling with strength of a thousand people
flutter; linger
heaving; pulling
tracer bullet; star shell; flare bomb
jinrikisha puller
building relocation (without disassembly); house relocation
airburst; bursting in the air (of a shell, projectile, etc.)
time-fuze shell; time-fuse shell; air-burst shell
tracer bullet; tracer shell
dragnet; seine
building relocation (without disassembly); house relocation
tugboat; tug; towboat
minohiki chicken (Japanese long-tailed breed)
曳 is thought to combine 日 (sun) and 乙 (a curved line), possibly depicting the sun being pulled or dragged across the sky. The exact historical development is uncertain.
The top 日 looks like a heavy object, and the lower 乙 resembles a hook or rope pulling it. Imagine dragging a stubborn sun across the sky with a rope.
For エイ, picture pulling a heavy 'A' frame with great effort: 'A' -> エイ, and you tug it into place.