Based on the rulebook context:

• When you gain a Wound, you draw a Wound card and place it in your player board’s leftmost empty Wound slot. The Wound may be face-up or face-down as specified (p.11).
• Stamina is spent to Sprint, and if your Stamina token moves onto a face-down Wound icon, you immediately gain a face-down Wound card (p.8).
• Investigators immediately die if they gain their 4th Wound, whether it is face-up or face-down (p.11).

Sequence of Effects

• The rules specify that if your Stamina token moves onto a face-down Wound icon, you immediately gain a face-down Wound (p.8).
• There is no rule stating that gaining a Wound causes you to lose Stamina; rather, losing Stamina can cause you to gain a Wound if you cross a face-down Wound icon on the Stamina track.

Timing and Simultaneity

• The Stamina loss occurs first (e.g., by Sprinting or another effect).
• If this Stamina loss causes your token to move onto a face-down Wound icon, you then immediately gain a face-down Wound.
• If this is your 4th Wound, you immediately die (p.11).

Could This Cause Simultaneous Death?

• The rules do not describe a situation where Stamina loss and Wound gain happen simultaneously; the Stamina loss triggers the Wound gain.
• Death occurs only when the 4th Wound is gained.

Summary

• Stamina loss occurs first. If this causes you to cross a face-down Wound icon, you then immediately gain a face-down Wound.
• If this is your 4th Wound, you die at that moment.
• There is no rule for simultaneous death from Stamina and Wounds; death is only checked upon gaining the 4th Wound.

Source: p.8, p.10, p.11

Source: Core Rules (pp. 10, 11, 24, 27, 8)