Concerning Norman Crawford's "premise that they don't have any historical connections is not reliable."

I once commented at Manchester to a leading teacher among the "tight OBs" that George Muller would not have gone along with what he was teaching. His reply was, "I don't go back to George Muller. He received unbaptised believers." I'm not sure whether Mr Muller did or he didn't, nor what was meant by "unbaptised believers." However, the first part of his reply lines up pretty much with Norman Crawford's account of the "assemblies" on the North American Continent. 

I suggest that those converted in the 1859 revival in Northern Ireland and Scotland found the "open brethren" the nearest best fit to their convictions, believer's baptism as a requisite for reception to the "local assembly" being one of theirs. That known Christians who had not left "the denominations" would be received to break bread among "the assemblies" was unacceptable in their view. 

Hence, one reason for the "tightening process" by their influence among many OB gatherings. A notion developed was of many distinctions between "the church, the body" and "an assembly" (Norman Crawford listed 27 in one of his books, though he thought there might be as many as 40!), and that being in "an assembly" was distinct from being in "the body."

A Christian must join a local assembly and be one of its members before being permitted to break bread. He must learn the truth of the assembly, because being "unlearned" he does not know that God is among them! 

There is no universal Christian fellowship, just "local assembly fellowship," and the partaking of the one loaf at the Lord's supper expresses the unity of the local assembly in which they gather, not the whole body of Christ on earth. Indeed, "the body of Christ is not on earth." There is no universal church, no church on earth, just local assemblies! 

(Incidentally, before anyone jumps to conclusions, I do not believe in "open reception" - the expression "known Christians" above meaning those known to be sound in doctrine, manner of life, and associations.)