String analysis is the problem of reasoning about how strings are manipulated by a program. It has numerous applications including automatic detection of cross-site scripting, and automatic test-case generation. A popular string analysis technique includes symbolic executions, which at their core use constraint solvers over string domains, a.k.a. string solvers. Such solvers typically reason about constraints expressed in theories over strings with the concatenation operator as an atomic constraint. In recent years, researchers started to recognise the importance of incorporating the replace-all operator (i.e. replace all occurrences of a string by another string) and, more generally, finite-state transductions in the theories of strings with concatenation. Such string operations are typically crucial for reasoning about XSS vulnerabilities in web applications, especially for modelling sanitisation functions and implicit browser transductions (e.g. innerHTML). Although this results in an undecidable theory in general, it was recently shown that the straight-line fragment of the theory is decidable, and is sufficiently expressive in practice for many applications. In this paper, we provide the first string solver that can reason about constraints involving both concatenation and finite-state transductions, and that is a decision procedure for several relevant fragments, including straight-line. The main challenge that we address in the paper is the prohibitive worst-case computational complexity of the theory (double-exponential time), which is exponentially harder than the case without finite-state transductions. To this end, we propose a method that exploits succinct alternating finite-state automata as concise symbolic representations of string constraints. Compared to methods that use representations based on nondeterministic machines, alternation offers not only (expected) exponential savings in space when representing Boolean combinations of transducers, but, importantly, also a possibility of succinct representation of otherwise costly combinations of transducers and concatenation. Reasoning about the emptiness of the AFA language requires a state-space exploration in an exponential-sized graph. To this end, we use the model checking algorithms like IC3 for solving the problem. We have implemented our algorithm and demonstrated its efficacy on string benchmarks that are derived from cross-site scripting analysis and other examples in the literature.