Eq = x'[t] + (x[t] - λ) == 0;

WhenEvent[x'[t] == 0, λ -> x[t]].

λ = 1;
Eq = x'[t] + (x[t] - λ) == 0;
sol = NDSolve[{Eq, x[0] == 0,
               WhenEvent[x'[t] == 0., λ -> x[t]]
              }, x, {t, 0, 5}][[1]]

equation = x'[t] + (x[t] - λ[t]) == 0;
sol = NDSolveValue[{equation, x[0] == 0, λ[0] == 1,
   WhenEvent[x'[t] == 0.25, λ[t] -> x[t]]}, x, {t, 0, 5},
  DiscreteVariables -> {λ}]