I added a brief description of how transition paths work to the VFI Toolkit Pseudo-Codes pdf
I copy paste it here for convenience
We consider a transition path of length T time periods. Solving for a transition path general equilibrium, we are looking for some path-of-parameters that are determined in general equilibrium, call them \Theta such that our general equilibrium equations equal zero in every time period, GEeqn(\Theta)=0. Note that at some level this is the same thing we did for a stationary equilibrium, except that now \Theta=\{\theta_1,....,\theta_T\} is a time-path of general-eqm parameters, rather than just parameters at a single point in time.
There are various algorithms for doing this (\textit{heteroagentoptions.fminalgo} determines which is used), but all of them essentially boil down to the following,
\begin{algorithmic}
\State Guess some initial \Theta^0 (time-path on general-equilibrium parameters)
\State Define GeneralEqmConditions=\infty
\While GeneralEqmConditions>tolerance
\State Solve backward, t=T-1,\dots,2,1 for the value fn and policy at each t (given current \theta^k_t)
\State Solve forward, t=1,2,\dots,T for the agent distribution (given current \theta^k_t and Policy_t we just found)
\State Solve the aggregate variables at each t (given current \theta_t^k and policy/agent dist we just found for t)
\State Evaluate GeneralEqmConditions=[GEeqn_1(AggVars_1,\theta_1^k), $GEeqn_1(AggVars_1,\theta_1^k),$$\dots,$ GEeqn_T(AggVars_T,\theta_T^k)] (evaluate general eqm eqns at each time period)
\State Update \Theta^{k+1} (in some way, likely based on \Theta^k and GeneralEqmConditions)
\EndWhile
\ \Return \theta
\end{algorithmic}
The difference between the different \textit{fminalgo} is how we update the \Theta. The most common way to update is as a shooting algorithm.
In a stationary equilibrium, parameters other than those parameters which were being determined in general equilibrium (the general equilibrium parameters) were all constant (in time) and kept in Parameters structure. When we solve transition paths, some parameters might vary over time, and this is done with the ParamPath (a path on exogenous parameters), while any other parameters not in this path are assumed to be constant at their values in Parameters structure. It is not described in algorithm above, but obviously we have to make sure to use the correct parameters in each time period.
It is important to solve backward for policy and forward for agent distribution. For convenience, aggregate variables and general eqm conditions are often solved forwards while doing agent distribution, but in principle how we deal with these (in terms of timing) is not important is most models (can be important if we use lagged/leading parameter values for something).
Whether the problem is finite or infinite-horizon value functions, and whether or not there are permanent types of different kinds of shocks is unimportant, in the sense that the above steps are all the same steps, just that the details of their implementation is changed.
PS. This looks a lot like how stationary general eqm is solved (for \theta^k, solve V and Policy, solve agent dist, evaluate agg vars, evaluate general eqm eqns, update \theta^{k+1}), just that now everything is a path instead of a value at a single point in time.
