What a colourful crossword! The completed grid looks like this:

The answers to the clues are:

Across: PER, BRB, BEL, CIV, ITA, CMR, GTM, FRA, CAN, SEN, MEX
Down: ARM, LBY, EST, AUT, GAB, NLD, SLE, MMR, RUS, BGR, AZE, LTU

(Where the clue is spelled backwards, the answer must also be backwards.)

These are ...

... the ISO 3166-1 country codes for Peru, Barbados, Belgium, Côte d'Ivoire, Italy, Cameroon, Guatemala, France, Canada, Senegal, Mexico and Armenia, Libya, Estonia, Austria, Gabon, the Netherlands, Sierra Leona Mynmar, Russia, Bulgaria, Azerbaijan, Lithuania.

The flags of all countries in the across clues are vertically striped tricolours; the flags of the countries in the down clues are hoizontally striped tricolours. So we must enter the colours of these flags into the grid. (Coats of arms and stars and such are ignored.)

That explains why ...

... the empty grids aren't white, but light grey: White is one of the colours that can be filled in.

There are two hints in the title and preamble:

To flag can mean to get tired and to flag something means to draw one's attention to it. (But I didn't see these hints until Stiv pointed out their existence in a comment.)

What a colourful crossword! The completed grid looks like this:

The answers to the clues are:

Across: PER<, BRB, BEL, CIV, ITA<, CMR, GTM<, FRA<, CAN, SEN<, MEX<
Down: ARM, LBY<, EST, AUT, GAB, NLD, SLE<, MMR, RUS<, BGR<, AZE, LTU

(Where the clue is spelled backwards (<), the answer must also be backwards.)

These are ...

... the ISO 3166-1 country codes for Peru, Barbados, Belgium, Côte d'Ivoire, Italy, Cameroon, Guatemala, France, Canada, Senegal, Mexico and Armenia, Libya, Estonia, Austria, Gabon, the Netherlands, Sierra Leone, Myanmar, Russia, Bulgaria, Azerbaijan, Lithuania.

The flags of all countries in the across clues are vertically striped tricolours; the flags of the countries in the down clues are horizontally striped tricolours. So we must enter the colours of these flags into the grid. (Coats of arms and stars and such are ignored.)

That explains why ...

... the empty grids aren't white, but light grey: White is one of the colours that can be filled in.

There are two hints in the title and preamble:

To flag can mean to get tired and to flag something means to draw one's attention to it. (But I didn't see these hints until Stiv pointed out their existence in a comment.)

What a colourful crossword! The completed grid looks like this:

The answers to the clues are:

Across: PER, BRB, BEL, CIV, ITA, CMR, GTM, FRA, CAN, SEN, MEX
Down: ARM, LBY, EST, AUT, GAB, NLD, SLE, MMR, RUS, BGR, AZE, LTU

(Where the clue is spelled backwards, the answer must also be backwards.)

These are ...

... the ISO 3166-1 country codes for Peru, Barbados, Belgium, Côte d'Ivoire, Italy, Cameroon, Guatemala, France, Canada, Senegal, Mexico and Armenia, Libya, Estonia, Austria, Gabon, the Netherlands, Sierra Leona Mynmar, Russia, Bulgaria, Azerbaijan, Lithuania.

The flags of all countries in the across clues are vertically striped tricolours; the flags of the countries in the down clues are hoizontally striped tricolours. So we must enter the colours of these flags into the grid. (Coats of arms and stars and such are ignored.)

That explains why ...

... the empty grids aren't white, but light grey: White is one of the colours that can be filled in.

What a colourful crossword! The completed grid looks like this:

The answers to the clues are:

Across: PER, BRB, BEL, CIV, ITA, CMR, GTM, FRA, CAN, SEN, MEX
Down: ARM, LBY, EST, AUT, GAB, NLD, SLE, MMR, RUS, BGR, AZE, LTU

(Where the clue is spelled backwards, the answer must also be backwards.)

These are ...

... the ISO 3166-1 country codes for Peru, Barbados, Belgium, Côte d'Ivoire, Italy, Cameroon, Guatemala, France, Canada, Senegal, Mexico and Armenia, Libya, Estonia, Austria, Gabon, the Netherlands, Sierra Leona Mynmar, Russia, Bulgaria, Azerbaijan, Lithuania.

The flags of all countries in the across clues are vertically striped tricolours; the flags of the countries in the down clues are hoizontally striped tricolours. So we must enter the colours of these flags into the grid. (Coats of arms and stars and such are ignored.)

That explains why ...

... the empty grids aren't white, but light grey: White is one of the colours that can be filled in.

There are two hints in the title and preamble:

To flag can mean to get tired and to flag something means to draw one's attention to it. (But I didn't see these hints until Stiv pointed out their existence in a comment.)