We consider a multistage framework introduced recently (Eisenstat et al., Gupta et al., both in ICALP 2014), where given a time horizon t= 1, 2, …, T, the input is a sequence of instances of a (static) combinatorial optimization problem I1, I2, …, IT, (one for each time step), and the goal is to find a sequence of solutions S1, S2, …, ST (one for each time step) reaching a tradeoff between the quality of the solutions in each time step and the stability/similarity of the solutions in consecutive time steps. For several polynomial-time solvable problems, such as Minimum Cost Perfect Matching, the multistage variant becomes hard to approximate (even for two time steps for Minimum Cost Perfect Matching). In this paper, we study multistage variants of some important discrete minimization problems (including Minimum Cut, Vertex Cover, Prize-Collecting Steiner Tree, Prize-Collecting Traveling Salesman). We focus on the natural question of whether linear-programming-based methods may help in developing good approximation algorithms in this framework. We first introduce a new two-threshold rounding scheme, tailored for multistage problems. Using this rounding scheme we propose a 3.53-approximation algorithm for a multistage variant of the Prize-Collecting Steiner Tree problem, and a 3.034-approximation algorithm for a multistage variant of the Prize-Collecting Metric Traveling Salesman problem. Then, we show that Min Cut remains polytime solvable in its multistage variant, and Vertex Cover remains 2-approximable, as particular cases of a more general statement which easily follows from the work of (Hochbaum, EJOR 2002) on monotone and IP2 problems.