  We present detailed magnetometry and muon-spin rotation data on
polycrystalline samples of overdoped, non-superconducting LiFe$_{1-x}$Ni$_x$As
($x = 0.1,\,0.2$) and Li$_{1-y}$Fe$_{1+y}$As ($0\leq y\leq 0.04$) as well as
superconducting LiFeAs. While LiFe$_{1-x}$Ni$_x$As exhibits weak
antiferromagnetic fluctuations down to $1.5\,{\rm K}$, Li$_{1-y}$Fe$_{1+y}$As
samples, which have a much smaller deviation from the $1:1:1$ stoichiometry,
show a crossover from ferromagnetic to antiferromagnetic fluctuations on
cooling and a freezing of dynamically fluctuating moments at low temperatures.
We do not find any signatures of time-reversal symmetry breaking in
stoichiometric LiFeAs that would support recent predictions of triplet pairing.
