We study the fundamental issue of decidability of satisfiability over string logics with concatenations and finite-state transducers as atomic operations. Although restricting to one type of operations yields decidability, little is known about the decidability of their combined theory, which is especially relevant when analysing security vulnerabilities of dynamic web pages in a more realistic browser model. On the one hand, word equations (string logic with concatenations) cannot precisely capture sanitisation functions (e.g. htmlescape) and implicit browser transductions (e.g. innerHTML mutations). On the other hand, transducers suffer from the reverse problem of being able to model sanitisation functions and browser transductions, but not string concatenations. Naively combining word equations and transducers easily leads to an undecidable logic. Our main contribution is to show that the straight-line fragment of the logic is decidable (complexity ranges from PSPACE to EXPSPACE). The fragment can express the program logics of straight-line string-manipulating programs with concatenations and transductions as atomic operations, which arise when performing bounded model checking or dynamic symbolic executions. We demonstrate that the logic can naturally express constraints required for analysing mutation XSS in web applications. Finally, we show that the logic remains decidable in the presence of length, letter-counting, regular, and indexOf constraints.