```python
from typing import List


def lay_conveyor(floor: List[str], row: int, col: int, length: int) -> List[str]:
    if row < 0 or row >= len(floor):
        raise ValueError("the run starts off the plan")
    cells = list(floor[row])
    if col + length > len(cells):
        raise ValueError("the run would pass the last column")
    for i in range(length):
        if cells[col + i] != ".":
            raise ValueError("cell covered by the run is not open")
    floor[row] = "".join(["="] * length) + cells[col + length:]
    return floor
```