I would like to use Double Time feat on a flying figure which is NOT a transporter.

DOUBLE-TIME prerequisite:
Transporter or 'boot' speed mode.

Can I therefore assign this feat to a flying figure which gains Earthbound later on the dial?

EARTHBOUND:
This character is grounded and has the boot speed mode instead of its printed speed mode. If this character is soaring when it gains Earthbound, this character is dealt 1 damage. This power may not be countered.

If I can, can I also assign Double Time feat to a flying figure which does not gain Earthbound but instead the BFC Crosswinds is played?

CROSSWINDS:
Characters with the wing speed symbol gain Eathbound. This ability may not be countered.

Thanks for your help.

You can assign Double Time to a hoverer that has the Earthbound power printed on his dial. You cannot assign it to a hoverer without Earthbound, and since Crosswinds doesn't activate until after the game starts, if your hoverer doesn't have EB on his dial, he can't be assigned the Feat in the first place.

Be carefull when using double-time on some Earthbound fliers...especially those with it on their last click! For Captain Atom is should be swell.