// Using currently available APIs
List<Task<int>> downloadTasks = downloadTasksQuery.ToList();
while (downloadTasks.Any())
{
    Task<int> finishedTask = await Task.WhenAny(downloadTasks);
    downloadTasks.Remove(finishedTask);
    Process(await finishedTask);
}

namespace System.Threading.Tasks
{
    public class Task : IAsyncResult, IDisposable
    {
        public static IAsyncEnumerable<Task> WhenEach(params Task[] tasks);  // Please change the name, if needed
        public static IAsyncEnumerable<Task> WhenEach(IEnumerable<Task> tasks);
        public static IAsyncEnumerable<Task<TResult>> WhenEach(params Task<TResult>[] tasks); 
        public static IAsyncEnumerable<Task<TResult>> WhenEach(IEnumerable<Task<TResult>> tasks);
    }
}

// Using newly created APIs
await foreach (var finishedTask in Task.WhenEach(downloadTasksQuery))
{
    Process(await finishedTask);
}