Four distinct lines $L_1,L_2,L_3,L_4$ are given in the plane: $L_1$ and $L_2$ are respectively parallel to $L_3$ and $L_4$. Find the locus of a point moving so that the sum of its perpendicular distances from the four lines is constant.
1. **Define the problem and setup the coordinate system:**
   We are given four distinct lines \( L_1, L_2, L_3, L_4 \) in the plane such that \( L_1 \parallel L_3 \) and \( L_2 \parallel L_4 \). We need to find the locus of a point \( P \) such that the sum of its perpendicular distances from these four lines is constant.

2. **Define the distances between the parallel lines:**
   Let \( x \) be the perpendicular distance between \( L_1 \) and \( L_3 \), and \( y \) be the perpendicular distance between \( L_2 \) and \( L_4 \).

3. **Consider the sum of distances from a point \( P \) to the lines:**
   Let \( d_1, d_2, d_3, d_4 \) be the perpendicular distances from \( P \) to \( L_1, L_2, L_3, L_4 \) respectively. We are given that \( d_1 + d_2 + d_3 + d_4 = D \), where \( D \) is a constant.

4. **Analyze different cases based on the value of \( D \):**
   - **Case 1: \( D < x + y \)**
     - This case is impossible because the minimum sum of distances from any point to the four lines is \( x + y \). Therefore, there are no points \( P \) satisfying this condition. Hence, the locus is empty.

   - **Case 2: \( D = x + y \)**
     - In this case, the sum of distances is exactly \( x + y \). This is satisfied by all points inside and on the boundary of the parallelogram formed by the lines \( L_1, L_2, L_3, L_4 \). Therefore, the locus is the boundary and interior of the parallelogram.

   - **Case 3: \( D > x + y \)**
     - For \( D > x + y \), consider the geometric properties of the distances. The locus of points \( P \) such that the sum of distances to the four lines is greater than \( x + y \) forms a centrally symmetric octagon. This can be understood by considering the sum of distances from a point to the sides of an isosceles triangle, which remains constant as the point moves parallel to the base.

5. **Conclusion:**
   - For \( D < x + y \), the locus is empty.
   - For \( D = x + y \), the locus is the boundary and interior of the parallelogram formed by the lines.
   - For \( D > x + y \), the locus is a centrally symmetric octagon.

The final answer is the locus is a centrally symmetric octagon for \( \boxed{ D > x + y } \), the boundary and interior of the parallelogram for \( D = x + y \), and empty for \( D < x + y \).