The word walking refers to the subject of the sentence, not to the woman. If the writer wishes to make it refer to the woman, he must recast the sentence:
He saw a woman accompanied by two children, walking slowly down the road.
Participial phrases preceded by a conjunction or by a preposition, nouns in apposition, adjectives, and adjective phrases come under the same rule if they begin the sentence.
| On arriving in Chicago, his friends met him at the station. | When he arrived (or, On his arrival) in Chicago, his friends met him at the station. |
|---|---|
| A soldier of proved valor, they entrusted him with the defence of the city. | A soldier of proved valor, he was entrusted with the defence of the city. |
| Young and inexperienced, the task seemed easy to me. | Young and inexperienced, I thought the task easy. |
| Without a friend to counsel him, the temptation proved irresistible. | Without a friend to counsel him, he found the temptation irresistible. |
Sentences violating this rule are often ludicrous.
Being in a dilapidated condition, I was able to buy the house very cheap.
Wondering irresolutely what to do next, the clock struck twelve.