Precession wrt invariable plane

Is it true that when a body's position is expressed as longitude and
latitude in the invariable plane system, and then recalculated for a
different epoch (assuming no proper motion), the latitude remains
unchanged?

There's some intuitive sense in the idea, but it seems to me that the
conversion from standard (J2000 or apparent) ecliptic coordinates to
invariable-plane coordinates would have to be a little more complex
than what I have seen in textbooks - or else the inclination of the
inv.p. wrt the ecliptic is more complex.

Is there a place online where this is discussed? I have seen
tantalizing references, but no link to a substantive discussion.