Evaluate the following expression:
$$\frac{5}{8}\times 2\frac{3}{5}÷\frac{4}{9}=?$$

First, convert the mixed fraction into an improper fraction:
$$2\frac{3}{5}=\frac{2\times 5+3}{5}=\frac{13}{5}$$

Now substitute it back into the expression:

$$\frac{5}{8}\times \frac{13}{5}÷\frac{4}{9}$$

According to the order of operations, division by a fraction is equivalent to multiplication by its reciprocal:

$$\frac{5}{8}\times \frac{13}{5}\times \frac{9}{4}$$

Cancel the common factor $5$ in the numerator and denominator:

$$=\frac{13\times 9}{8\times 4}=\frac{117}{32}$$

Convert the improper fraction back into a mixed fraction:

$$\frac{117}{32}=3\frac{21}{32}$$