As the capacity and reliability of mobile networks increases, so does the demand for more responsive end-to-end services: applications such as augmented reality, live video conferencing, and smart or autonomous vehicles require reliable, throughput-intensive end-to-end communications with strict delay constraints. Only consistently reliable delivery of data flows well within human interactivity deadlines will enable a truly immersive user experience. To enable data delivery within pre-defined deadlines, controlled on demand by an application or its user, we propose and demonstrate a novel transport-layer protocol for explicit latency control called latency-controlled end-to-end aggregation protocol (LEAP). The LEAP splits a data flow with quality of service (QoS) constraints into multiple subflows that are delivered over multiple parallel links (e.g., Wi-Fi and LTE in a standard smartphone, WiGig, and 5G in the near future). The subflow data rates are set based on a novel proactive forecasting of the achievable channel capacity, subject to application-specific QoS constraints. Cross-path encoding and redundancy adaptation are then used to deliberately balance the trade-off between maximum throughput, required delay, and minimum reliability as function of application/user-specific input parameters. When compared to leading state-of-the-art transport protocols in live network experiments, LEAP exhibits a superior capacity to reliably provide a high and stable throughput with bounded latency, both in wired and wireless scenarios. The LEAP is also the first protocol to allow applications to explicitly set their priorities, giving them the freedom to set the operating point in the trade-off between throughput, latency, and reliability.