The methods of achieving this objective could be varied, including - but not limited to:
1) direct payments from the satalite (or colony or subsidiary) to the imperial treasury (or parent company) for a variety of reasons (some more legitimate than others) from simple payments by request or under a variety of other legal forms such as "reimbursements" or "royalty" or "licence" payments,
2) "pushing" expenses from the imperial power (or parent company) "down" into the satalite (or colony or subsidiary) that otherwise might have been borne by the imperial power (or parent company).
3) through trade, either or both of the forms:
3-a) trade where goods from the satelite (or colony or subsidiary) would be sold at a price favourable to the imperial power (or parent company) so that the profit on the ultimate sale of the goods would be "captured" by the imperial power (or parent company)
3-b) trade where good from the imperial power (or parent company) would be sold to the satelite (or subsidiary company) at a price favorable to the imperial power (the reverse of 3-a),
together these are in modern language "a transfer pricing strategy". The benefits of such trade might not have gone only to the treasury but also to people allied with the political elite.
4) interest (or dividends) on any capital "invested" in the satelite (or colony or subsidiary) via direct loans or whatever means.
There may be other "inventive" methods used at the time (as opposed to more modern methods) and could have including "forcing" the satalite to take loans from the market in order to pay any amounts due to the imperial power (or parent company).
The modern approach (in the absence of colonies, this applies to multi-national companies) would be to achieve a "sustainable" structure - in other words, a form of payments that would continue to operate into the future.
At the time, the "sustainable" concept seems to have been limited to simply avoiding a revolt against the imperial power - it was a balancing act.
Napoleon did this as did England, Russia, Austria, Spain, Portugal, the Ottoman Empire or any other power of the period.
The French Emperor's methods tended to be more "direct" than those of, say, England.
Britain neglected its continental allies during 1806 and early 1807 as they sent ships and men to gather the benefits of the Cape or South America and to cover a potential revolt in India. They did this to guard "their" trade (actually Napoleon's Continental System started to work - Britain needed new markets so the "unauthorized" move against Buenos Aires was "accepted"; the key error of the Continental System was Napoleon's blindness to the trade available to England from the Spanish and Portuguese colonies in America).
These English methods were "indirect"; still, the objective was the same as Napoleon's "milking". - R