The West Indian Snipe was described in 2016, its fossil or subfossil remains were recovered from late Quaternary depostits on several islands including the Bahamas, the Cayman Islands, Cuba, and the Turks and Caicos Islands.
The West Indian Snipe was a volant species, but probably had quite short wings, this can be assumed from the shape of its wingbones. 
The species probably died out for more or less natural reasons, following changing environmental conditions due to changes in the height of the sea level after the beginning of the Holocene. The last remaining populations were then probably extirpated by the first Amerindian settlers. ********************
 David W. Steadman; Oona M. Takano: A new extinct species of Snipe (Aves: Scolopacidae: Gallinago) from the West Indies. Zootaxa 4109(3): 345-358. 2016
The Christmas Sandpiper was apparently restricted to the Kiritimati atoll in Kiribati when it was discovered, described and depicted during the third voyage under the command of Captain James Cook, which visited the atoll from 1777 to 1778; it might very well have once been much wider distributed in the archipelago.
The species disappeared due to the introduction of rats to its home, it might already have been on the brink of extinction when it was first met by Europeans and probably disappeared shortly after.
This species was first recorded from a Holocene fossil site from Trouing Jean Paul, a high elevation limestone sinkhole in the Massif de la Selle, Haiti, which represents the prey remains of the the endemic Ashy-faced Barn Owl (Tyto glaucops (Kaup)). These can be dated to an age of about 1000 CE, thus date to a time when most of the larger endemic bird – and mammal species already had been extirpated by the first Amerindian settlers, but still some 500 years before the arrival of the first European conquerors. 
No woodcock species is today known to inhabit the Caribbean region, but formerly there appears to have been a small radiation of at least two species, maybe some more to be discovered in the future.
The Hispaniolan Woodcock most likely disapperaed due to hunting and habitat destruction, it may even have survived into quite historical times, but this assumption needs to be proven.
 David W. Steadman; Oona M. Takano: A late-Holocene bird community from Hispaniola: Refining the chronology of vertebrate extinction in the West Indies. Holocene 23(7): 936-944. 2013  Oona M. Takano; David W. Steadman: A new species of Woodcock (Aves: Scolopacidae: Scolopax) from Hispaniola, West Indies. Zootaxa 4032(1): 117-126. 2015
The Henderson Island Sandpiper is known from subfossil remains, which were compared to the bones of the extant Tuamotu Sandpiper (Prosobonia parvirostris (Peale)); it differed from that species by its longer leg bones and its reduced outer wing bones.
Given the fact that this species apparently had some reduced flight abilities, it might have indeed been restricted to Henderson Island. 
The species was finally described in 2020 as a new species, after having been known for almost 26 years. 
 Graham M. Wragg; Marshall I. Weisler: Extinctions and new records of birds from Henderson Island. Notornis 41: 61-70. 1994  Vanesa L. De Pietri; Trevor H. Worthy; R. Paul Scofield; Theresa L. Cole; Jamie R. Wood; Kieren J. Mitchell; Alice Cibois; Justin J. F. J. Jansen; Alan J. Cooper; Shaohong Feng; Wanjun Chen; Alan J. D. Tennyson; Graham M. Wragg: A new extinct species of Polynesian sandpiper (Charadriiformes: Scolopacidae: Prosobonia) from Henderson Island, Pitcairn Island Group, and the phylogenetic relationships of Prosobonia. Zoological Journal of the Linnean Society 20: 1-26. 2020
This species, finally described in 2013, was until then known only from subfossil bones, which were found together with the remains of numerous other species of animals in the deposits of several caves on the Pindai Peninsula.
The New Caledonian Snipe was the second largest species of its genus, its flight apparat was very well developed, thus it obviously was a very good flyer.
The species disappeared sometimes after the colonization of New Caledonia’s islands at about 1500 B.C.E..
 David W. Steadman: Extinction and Biogeography of Tropical Pacific Birds. University of Chicago Press 2006  Trevor H. Worthy; Atholl Anderson; Christophe Sand: An extinct Austral snipe (Aves: Coenocorypha) from New Caledonia. Emu 113(4): 383-393. 2013