To solve the expression \((-3) \div \left(\frac{3}{4}\right) \times \left(\frac{2}{7}\right) \times \left(-\frac{5}{16}\right) \div \left(\frac{5}{7}\right)\), let's break it down step by step.
Step 1: Rewrite the divisions as multiplications by the reciprocal.
\[ (-3) \div \left(\frac{3}{4}\right) = (-3) \times \left(\frac{4}{3}\right) \]
\[ -\frac{5}{16} \div \left(\frac{5}{7}\right) = -\frac{5}{16} \times \left(\frac{7}{5}\right) \]
Now the expression can be rewritten as:
\[ (-3) \times \left(\frac{4}{3}\right) \times \left(\frac{2}{7}\right) \times \left(-\frac{5}{16}\right) \times \left(\frac{7}{5}\right) \]
Step 2: Simplify the terms.
\[ (-3) \times \left(\frac{4}{3}\right) = -\frac{12}{3} = -4 \]
Now the expression simplifies to:
\[ -4 \times \left(\frac{2}{7}\right) \times \left(-\frac{5}{16}\right) \times \left(\frac{7}{5}\right) \]
Step 3: Simplify further.
Multiplying \(-4\) and \(-\frac{5}{16}\):
\[ -4 \times -\frac{5}{16} = \frac{20}{16} = \frac{5}{4} \]
Now we have:
\[ \frac{5}{4} \times \left(\frac{2}{7}\right) \times \left(\frac{7}{5}\right) \]
Step 4: Multiply the fractions.
Notice that \(\frac{7}{5}\) cancels with \(\frac{5}{4}\) when we multiply:
\[ \frac{5}{4} \times \frac{2}{7} \times \frac{7}{5} = \frac{5 \cdot 2 \cdot 7}{4 \cdot 7 \cdot 5} \]
The \(5\) cancels out:
\[ = \frac{2}{4} = \frac{1}{2} \]
Thus, the final answer is:
\[ \boxed{\frac{1}{2}} \]