How does the parent account for selling some of its subsidiary shares while retaining control?

What is the key accounting distinction between an ownership change that retains control and one that loses control?

When the parent's ownership in a subsidiary decreases (control retained), what happens to the acquisition differential?

Before obtaining control, how is a small equity investment that is later stepped up to control reported, and what happens at the step-up?

How is control and consolidated interest determined with indirect shareholdings (e.g., parent → subsidiary → sub-subsidiary)?

How are a subsidiary's preferred shares considered when allocating consolidated net income?

What happens to the parent's interest when a subsidiary issues additional shares to the public?