This paper shows the strong converse and the dispersion of memoryless channels with cost constraints and performs refined analysis of the third order term in the asymptotic expansion of the maximum achievable channel coding rate, showing that it is equal to $\frac 1 2 \frac {\log n}{n}$ in most cases of interest. The analysis is based on a non-asymptotic converse bound expressed in terms of the distribution of a random variable termed the $\mathsf b$-tilted information density, which plays a role similar to that of the $\mathsf d$-tilted information in lossy source coding. We also analyze the fundamental limits of lossy joint-source-channel coding over channels with cost constraints.