We analyze the finite-blocklength performance of lossy joint source-channel codes (JSCC) in an unknown-channel framework, where the true channel is unknown but the source distribution is known. We establish achievability results for mismatched-design JSCC, where the code design is based on a channel $Q_{Y|X}$ but deployed over a different channel $P_{Y|X}$. Our one-shot achievability bound allows for standard Borel alphabets for the source, reproduction, channel input and channel output. The subsequent block coding result based on the normal approximation applies to stationary memoryless sources and memoryless, possibly nonstationary channels under regularity and moment conditions. The achievability bound is given in terms of the rate-distortion and rate-dispersion functions, as well as two channel-dependent quantities that we call the mismatched-design rate and mismatched-design rate-dispersion. We use a family of Gibbs posteriors parameterized by a single scalar as decoder-side kernels, and the envelope of the corresponding achievable rates recovers the generalized mutual information. In the stationary matched setting covered by our assumptions, our result recovers the achievability part of Kostina and Verdú's 2013 Gaussian approximation result and improves its third-order term. For block erasure channels, channel mismatch incurs no first- or second-order asymptotic penalty. We then construct a channel-blind family of source-channel codes that is second-order universal over stationary block erasure channels. Our code construction uses Poisson functional representations of suitable conditional probability measures to produce the encoder and decoder outputs.

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