Final answer:
The Townshend Acts threatened colonial assemblies by stripping them of the 'power of the purse', funding royal officials independently, and establishing more authoritative controls, significantly undermining colonial self-governance and fomenting revolutionary sentiments.
Step-by-step explanation:
The Townshend Acts presented a significant threat to colonial assemblies by asserting British authority in new and far-reaching ways. The Townshend Acts placed duties on common goods and took control away from colonial legislatures, notably by using the revenue to pay royal governors, thus rendering local assemblies powerless over fiscal decisions. This series of measures undermined the political autonomy of the colonial assemblies which had previously influenced royal officials through their control over financial resources.
Importantly, the Townshend Acts sought to establish a distinction between internal and external taxes, imposing the latter in response to colonial resistance to the former. However, the Revenues Act component of the Townshend Acts cut directly into the power of colonial assemblies by ensuring that royal officials were no longer dependent on local authorities for their salaries. The Act removed the so-called 'power of the purse' that assemblies wielded over officials, sharply weakening their influence.
The Restraining Act, for instance, directly targeted the New York assembly and nullified its legislation. The establishment of the Board of Customs Commissioners and the subsequent Vice-Admiralty Courts created further layers of control, removing from local assemblages the capacity to hold British officials accountable, thereby escalating tensions and contributing significantly to colonial protest and eventual revolution.