(Theorem) If a particle is moving at a constant velocity relative to K, it's also moving at a constant velocity relative to K'.
Although this follows immediately from the law of inertia, let me demonstrate it differently and explain the rationale behind afterwards.
(Proof) Let P, Q, and M be points on the trajectory of the particle in K, such that d(P, M) = d(M, Q). Let P', Q', and M' denote the corresponding points on the trajectory in K', respectively. Due to symmetry reasons: (a) projecting P, Q, and M, each when the particle is passing by, onto the x'-, y'-, and z'-axes yields the same Δx', Δy', and Δz' values between P' and M' as it does between M' and Q'; (b) the respective Δt' values are the same too. So in K', the (average) velocity of the particle between P' and M' is equal to that between M' and Q'. Thus, due to continuity, the velocity of the particle is constant in K'.
The advantage of this proof is that it's valid not only for particles but also for any moving entity, including the tip of a ray of light, or even just a point-like state propagating through K.
Note: the proof does not exclude the possibility that Δt' = 0, i.e. that the speed in K' is "infinite".