Similar to Kan condition for simplicial sets, there are also Kan condition for simplicial manifolds, that is, we ask the horn projection $p^k_j: X_k \to Hom(\Lambda[k,j], X)$ to be a surjective submersion. If $p^k_j$ is only a submersion, then we say that the Kan condition is satisfied locally.
For example, the nerve of local Lie groups do not satisfy Kan condition but it only violate the surjectiveness, thus it satisfies local Kan condition.
Now the question is that I have not yet met a simplicial manifold in everyday life which violate even local Kan condition.
It's easy to find simplicial sets which is not Kan complex. And I was told it is easy to find a simplicial scheme which violate local Kan condition. But I don't know how to understand it in differential geometry.
Here is an example, given by J.Pridham: take a manifold M (e.g. a line $\mathbb{R}$), then something (dimensionally) smaller N (e.g. a point 1). Then you build yourself a semi-simplicial object: by N source + target to M, e.g. 1=>R by s(1)=a, t(1)=b. Then you complete the degeneracy, and obtain a simplicial object. $X_0=R$, $X_1=R \sqcup 1$, etc. Then the 1st Kan condition, namely s or t, are not submersion at the image of 1. because of 1. Namely, you can disturb a simplicial manifold by disjointly adding something small, then on this small thing, the s and t are not submersions.