In most cases the two constructions are parallel. In one you have a possessive + gerund, because a gerund is a noun. In the other you have an noun or objective pronoun + participle, because a participle is a modifier. This applies to your first two sets of examples.
Your third set is different. In the first sentence of the third set, "looking" is a present participle that is part of a progressive/continuous verb. "Investigating" is a participle modifying the noun "police". The second sentence in the third sentence is incorrect.
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