We consider lattice walks in $\mathbb{R}^k$ confined to the region $0 < x_1 < x_2 \ldots < x_k$ with fixed (but arbitrary) starting and end points. The walks are required to be "reflectable", that is, we assume that the number of paths can be counted using the reflection principle. The main result is an asymptotic formula for the total number of walks of length $n$ with fixed but arbitrary starting and end point for a general class of walks as the number $n$ of steps tends to infinity. As applications, we find the asymptotics for the number of $k$-non-crossing tangled diagrams on the set $\{1,2, \ldots,n\}$ as $n$ tends to infinity, and asymptotics for the number of $k$-vicious walkers subject to a wall restriction in the random turns model as well as in the lock step model. Asymptotics for all of these objects were either known only for certain special cases, or have only been partially determined.