Say armor gunner Jimmy Horse is taken as a bondsman in a Lyran raid against the Horses, a probe to test the new Clan in the IS. It is fought like a normal Trial and Jimmy recognizes he is a bondsman to officer whoever in the LCAF.
I don't see a Lyran officer putting a bondscord on Jimmy Horse. and I don't see the LCAF recognizing Jimmy Horse as a member of its military (bondsman or otherwise). The LCAF is a Spheroid military, not a Clan touman. The LCAF has no legal precedent or military tradition for integrating captured enemy soldiers (Clan or otherwise) into its regiments, nevertheless trusting them in the field. Jimmy Horse has become a POW, not a bondsman.
(That said, I'm sure someone will quote a rule-of-cool exception where some Spheroid military takes on some Clan hero as a bondsman in the fluff of a novel or some other reference. But I'd contend that makes no sense and should not be considered a standard or regular occurence, even if it's technically canon.)
Similarly, depending on his politics, prejudices, and preconceptions, Jimmy Horse may not recognize the authority of a barbaric, non-Clan, Spheroid power (especially those money-grubbing Elsies!) to make him a bondsman.
Does Jimmy try to become a Mechwarrior in the LCAF or stay a treadhead?
Same sort of deal for the Elemental bloodlines who become standard infantry?
I hate to say it as a Horse fan, but absent some very limited exceptions with specific factions, often only in certain time periods, I don't think Horse tankers and conventional infantry have opportunities to become bondsmen. As already described, most Spheroid and Periphery factions don't (or shouldn't) take bondsmen. And most Clan factions lack enough quality armor and infantry assets to desire these Horse bondsmen in their toumans -- they have enough of their own solahma and test-downs to fill these low-priority roles.
A few possible exceptions:
-- Steel Viper conventional infantry in all eras (Vipers use large numbers of non-solahma conventional infantry in their second-line units, especially their Fang and Legion clusters)
-- Wolf armor during the Dark Age (if the Wolves were willing to take on Lyran armor units, they'd certainly take Horse armor)
-- Clan-affiliated mercenaries in all eras (Wolf's Dragoons and the like took Jaguar bondsmen, understood how to use them, and lacked the Clan prejudice against armor and conventional infantry)
-- Desperate Clans near extinction (Jaguars during Serpent, Cats during whatever Combine/Dominion war, Spirits during WoR, etc.)
-- Society and Bandit Caste
There may be others, but I think they would be fairly limited exceptions.
Even the Bears have ample access to test-downs of their own -- it's hard to see the Bears using Horse bondsmen to fill their Phalanx clusters when so many of their own sibkin fail trials of position all the time.
Of course, whatever works for your own headcannon and game.