Some of us who consider ourselves "Landmark" simply hold the following:
Jesus built the very first NT church during His earthly ministry; local congregations like that one have existed ever since that time, as He promised - however few, however scattered, however differing among themselves in incidentals.
There is no more need to prove a lineal ancestry than there is to prove our descent from Adam - we have his characteristics. It is interesting, but not essential. Some insist on a "mother church;" some identifying themselves as "Landmark" insist that only self-constituted churches are valid! So many areas differ - style of worship (some are "high church," some "Bapticostal"), views on education, elements of other groups strong in the area where they are, reaction against such other groups, (e.g., the old issue of burial in "holy ground," or whether we may eat inside the same building where we worship, or use of instruments in worship, or . . . . You fill in the blanks!)
But all "Landmark" folks agree that the Bible, as God gave it, is a verbally inspired and final standard, and most will say that the NT, which is comparatively clear, interprets the OT, which is often difficult, and by completing the divine revelation, changes our form of worship from that of Tabernacle-Temple sacrifices to a Lord's day celebration of the gospel. (Although, there are "7th Day Baptists" who also believe in a succession and are pretty well "Landmark" in their views!) Some are Pre-mil, some A-, and B. H. Carroll was during his active ministry one of the strongest Post-millenialists, so some who still read his classic "Interpretation of the English Bible" are heavily influenced by it in that area as well. Some hold sovereignty (election, predestination); some fight strongly against it. Many will not receive one another's baptisms! Yet all fit the general term because they fall somewhere within Major B's statements or my brief summary. Best- Charles Blair - Rom. 8:28