In Weibel's K-Book, before defining the lambda-operations on higher (Quillen) K-theory, he states that "Although many constructions of $\lambda$-operations have been proposed in more exotic settings, we shall restrict our attention in this section to operations defined using the $+$-construction."
I would actually very much like to hear about those more exotic settings. Could you point me to references which describe various general constructions of $\lambda$-operations in (various flavour of) K-theory? The more general the better.
See https://faculty.math.illinois.edu/~dan/cv.xhtml#extpow for a paper of mine and a link to one of Thomas Gunnarsson and Roland Schwänzl.