Pivoting from the Unitarians, and looking forward to Maundy Thursday. I'll go into the Universalist laws of fellowship (and how they changed) later, but suffice it to say now that state conventions, parishes and ministers were subject to them or risk losing their standing. For a few decades, at least, one of these laws concerned who could be admitted to the Lord's table, or Communion.
From the 1946 Laws of Fellowship
In every church the Communion of the Lord's Supper shall be statedly observed at such times as the laws thereof prescribe; and at every such service all persons present, whether members or not, who may feel it to be a duty or privilege to do so, shall be invited to participate.
This formulation goes back at least as far as 1891. It also appears in the 1951 version, but disappears in the next (1953) version when, with other specific rules related to Christianity, it was removed. (As for the reference to church laws, even today Universalist National Memorial Church, Washington requires it on "Thursday of Holy Week and at such other times as the Pastor and Diaconate may determine. At every such service all present shall be invited to partake.")
The reading of the law matches what is printed as an invitation to communion in the "red hymnal" Hymns of the Spirit service for communion, even though it was a joint Unitarian-Universalist production:
A Communion Service will be held in this Church at (stating the time). It is a service of commemoration, consecration and fellowship, open to all who desire to take part in it.
Interestingly, no such preface exists for the Communion service before the last solely Universalist hymnal, Hymns of the Church.