While more can always be said on the topic, I hope my brief summary in the following link might offer some basic ideas: Wikipedia alleges that contingents from Armenian Cilicia, Georgia and the Principality of Antioch were present at the sack of Baghdad by the Ilkhanate in 1258, accompanied by a big fat [citation needed]. I presume this is complete BS, but is there some slight grain of truth to it, even if not the absolute specifics?

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• Grigor of Akanc (d. 1335) mentions the cavalry of the Georgians and the Armenians, but I'm not sure whether his account can be regarded as contemporary.
• As is similar with the case of Russian principalities, succession strife (divided factions) played an important role for these Christian powers in submitting at least either of the claimant factions to the newly emerging Mongol Empire - in order to get support.