This work is concerned with the global continuation for solutions (λ,u,ξ) ∈ R × C1{0}([0,∞), RN) × Rk of the following system of ordinary differential equations: where F: [0,∞) × RN × U × J → RN and φ: U × J → X1, for some open sets J ⊂ R and U ⊂ Rk, and where RN = X1 ⊕ X2 is a given decomposition, with associated projection P: RN → X1. This problem gives rise to a nonlinear operator whose zeros correspond exactly to the solutions of the original problem. We give conditions on F and φ ensuring that this operator has the Fredholm property and is proper on the closed bounded subsets of R × C1{0}([0,∞), RN) × Rk. These conditions generalize to a parameter dependent situation some recent results obtained by Morris [24]. Under these assumptions, we study the global behavior of a particular connected set of solutions using a degree theory available for such operators and obtain global continuation theorems. In the second part of this work, we use our general results to prove the existence of solutions for the so-called swirling flow problem in fluid dynamics, which can be written as a system of two ordinary differential equations on the half-line together with boundary conditions. Having obtained a priori bounds on possible solutions to this problem, we are able to recover the existence result obtained by Mcleod [22] using ad hoc arguments. In the last part, we present some numerical computations and give pictures of solutions to this problem.