Henred
As far as I know, there is no specific history concerning the contingents of the 57th Coy., Imperial Yeomanry. Nor, for that matter, for the 15th Bn., I.Y. The medal roll for the 57th Coy., I.Y. (WO100/126) shows Capt. F. Lane-Fox as entitled to clasps "Cape Colony", "Orange Free State" and "South Africa 1901" and there is a marginal endorsement that he served with the "old Yeomanry)"; i.e., the first contingent. Capt. Lane-Fox's entry in "Boer War Services of Officers" confirms his time in SA (1900-1901) and three clasps and also shows him as having the rank of Honorary Captain in the Army and lately in the militia - presumably after the war.
Researching the activities of the 57th Coy., I.Y., or any IY Coy. for that matter, can be difficult and time consuming Sometimes the Coys. acted as part of the Battalion and sometimes alone. A place to start, however, is in the relevant pages of the SAFF Casualty roll; which generally shows the date and place of field casualties. Unfortunately, any entries for the 57th Coy. I.Y. are shown as part of the 15th I.Y. Bn. - but, using the medal roll against the SAFF entries - members of the 57th Coy., I.Y. could be isolated. Thus, a rough idea of the trekking of the 57th Coy. can be had.
Now, most of the above is a lot of work - and I have followed that path in researching the medals of I.Y. chaps in my collection. Not to any startling conclusion - but to my satisfaction as a medal collector. Deeper research about the 57th Coy. or the 15th Bn., I.Y. can be conducted in The Times History (Amery) or the Official History (Maurice). Both accessible in the Books section of this site. An excellent account of the formation of the Imperial Yeomanry and the later contingents may be found in "Absent Minded Beggars" by Will Bennett.
Good luck with your quest and the book about the Officer's Servant/batman.
Regards
IL.