After. The qualifying relationship must exist on the filing date, which means the U.S. entity must be owned by the petitioner and be doing business, or at least have secured premises under the new-office rules. A petition filed on a signed but unclosed purchase agreement describes a relationship that does not yet exist.
Close, register, then petition
Hypothetical example: a Haldimand County elevator company has an accepted offer on a U.S. mill with closing set in six weeks, and wants the manager on site the day after closing. The sequence cannot be shortened by filing early.
On the petition date the U.S. entity must exist, be owned by the Canadian company or its owners in a qualifying proportion, and either be operating or hold secured premises with a credible plan. Use the six weeks to prepare everything else: the manager's payroll history, the organizational chart, the job description in managerial terms, and the corporate documents.
File the day after closing with premium processing if the timeline is tight. In the meantime, the manager can attend the closing and meet staff as a business visitor, but she cannot perform productive work at the mill until the L-1A is approved. Build the operating plan around a short interim managed by existing U.S.
staff rather than around an approval date nobody can promise.