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 mismatched-design achievability result allows nonstationary channel laws and arbitrary standard Borel alphabets for the source, reproduction, channel input and channel output. 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. For block erasure channels, our result shows that channel mismatch incurs no penalty. We then show a second-order universal family of source-channel codes over the set of block erasure channels. Our code construction uses Poisson functional representations of suitable conditional probability measures to produce the encoder and decoder outputs. We use a parameterized family of Gibbs posteriors as the decoder-side kernels, whose envelope recovers the generalized mutual information.