And pulsars spin rates slowly change over time do to energy losses.
But I don't think these things would cripple a pulsar-based navigation system. Presumably any network of pulsars used for this purpose would be regularly monitored and timing solutions would be updated regularly. The updated timing solutions could be transmitted as-needed.
At any rate, a robust navigation network of pulsars would need a sufficient number of sources to guard against glitches breaking the navigation system.