secure "related-key" stream cipher Consider a stream cipher $E(k)$ which takes a key $k$ to produce a pseudo random keystream.

It should have the property that you can easily generate a set of at least 3 different keys $\{k_m, k_0, k_1, \cdots\}$ such that the key stream generated by the master key $E(k_m)$ is the same as XOR-ing all key streams of the other keys $E(k_0) \oplus E(k_1) \oplus \cdots$

Is this property a security risk by itself, or could such a cipher be used to speed up an encryption scheme with nested encryption, like onion routing?

If it could be secure, are there already any ciphers with such a property?
If you consider symmetric encryption schemes instead of just stream ciphers, you're looking for "key-homomorphic encryption". There has been some work on lattice-based fully key homomorphic encryption schemes (see here).