The Royal Sussex Regiment Brighton War Memorial notes the following timeline of deaths and activities:
Barttelot, Capt. Sir Walter George: killed in action at Retief's Nek, July 23rd, 1900 (1st battalion)
Bleach, Pvt. Frank - died at the Hospital, Bloemfontein, of enteric fever, on 14th March 1901.
Brookshaw, Pvt. Benjamin A - At Krugersdorp, with Dickinson's column August 1901.
Du Moulin, Lt. Col. Louis Eugene - was killed in action at Abrahan's Kraal, near Koffyfontein, January 28th, 1902. In February 1900, it proceeded to South Africa and then took part in the march from Bloemfontein to Pretoria, inclduing the engagements at Welkom Farm, Zand River and Doorn Kop, the occupations of Johannesberg and the engagement at Diamond Hill. Lieutenant du Molin was also present at the engagements round Bethlehem and in the Caledon Valley, and at Keitef's Nek and the surrender of Boer forces at Golden Gate August 1st, 1900. He also took part in the operations around round Thaba N'chu, Winburg, and LIndley, and in the engagements at Bothaville and Ventersburg. At one period of the war he was in command of a column and was mentioned in despatahces [London Gazette July 9th and September 10th, 1901] was promoted lieutenant-colonel and granted the medal with four clasps for his good services.
Gaston, Pvt. E. - Died 28 January 1902 at Leeuw River North
Maguire, Capt. Charles James Kinahan - killed in action at Diamond Hill, near Pretoria, June 11th, 1900
The U.K. National Archive records for the Royal Sussex Regiment notes the following about its activities during the Boer War:
The South African War, 1899-1902
During the Boer War of 1899-1902 the 1st Battalion was reinforced by the Militia and by three successive companies formed from the Volunteer units of the Regiment. The Royal Sussex Light Infantry Militia, which was first embodied in 1778, was reconstituted as the 3rd Battalion, The Royal Sussex Regiment, in 1881, and served in South Africa from April 1901. The volunteer movement, which can be traced back at least to the Napoleonic War period, was revived in 1859, on fears of war with France, and local corps of rifle volunteers were established in towns throughout the county. The movement was reorganised in 1887 with the formation of the 1st and 2nd Volunteer Battalions, and it became more closely allied to the Regiment. Both battalions supplied drafts for the Volunteer Active Service Companies which were attached to the 1st Battalion, The Royal Sussex Regiment, during the South African War. This was the first time that the Volunteers had fought for their country overseas. During the latter part of the War members of the Royal Sussex contingent were mounted and formed into mobile columns to round up the Boer commandos
Finally, by way of preserving a comment on the question courtesy of @Comintern, here is a basic regimental history for the Royal Sussex during the Second Boer War