Partial answer
I've got the countries figured out, but there's five of them (six including Kiribati), not three:
Big flag on the left: KIRIBATI (I haven't even spoilered this since it's mentioned in the question).
Flags on the right: AZERBAIJAN and NETHERLANDS (internet domains .AZ and .NL, calling codes 994 and 31, if that matters).
Mottos below: TUVALU (national motto "Tuvalu mo te Atua" = "Tuvalu for the Almighty"), ??? (I couldn't find any national motto or anthem with the words "piety" and "trust", although it could be a non-standard translation of something), TURKMENISTAN (national motto "Türkmenistan Bitaraplygyň watanydyr" = "Turkmenistan is the motherland of Neutrality").
Big flag on the left:
KIRIBATI (I haven't even spoilered this since it's mentioned in the question).
Flags on the right:
AZERBAIJAN (internet domain .AZ, calling code 994)
NETHERLANDS (internet domain .NL, calling code 31)Mottos below:
TUVALU (internet domain .TV, calling code 688). National motto "Tuvalu mo te Atua" = "Tuvalu for the Almighty".
SURINAME (internet domain .SR, calling code 597). National motto "Justitia – Pietas – Fides" = "Justice – Piety – Trust". Interesting that only the last two-thirds of this is included; that must mean something, but I'm not sure what.
TURKMENISTAN (internet domain .TM, calling code 993). National motto "Türkmenistan Bitaraplygyň watanydyr" = "Turkmenistan is the motherland of Neutrality".
Possible method of answering (especially given the title):
something to do with the two-letter internet domains for these countries. Interestingly, Kiribati (.KI) is exactly a 3-letter Caesar shift from the Netherlands (.NL), but between them is not Azerbaijan but Macedonia (.MK). Meanwhile, Tuvalu is .TV and Turkmenistan is .TM.
By the way,
what I've got as Tuvalu could also be indicating TOKELAU (motto "Tokelau mo te Atua", but a territory of New Zealand rather than an independent state).