Is the deformation limit of Ricci-flat Kahler manifolds Kahler?

There are counterexamples: a Moishezon manifold, which has trivial canonical class and is birationally equivalent to a hyperkahler manifold, is also deformationally equivalent to a hyperkaehler manifold (this is a result of Huybrechts, I am not sure if he stated it in this generality, but his proof certainly works). Such Moishezon manifolds can be non-Kaehler (see e.g. Periods of Enriques Manifolds, Keiji Oguiso, Stefan Schroeer, http://arxiv.org/abs/1010.0820, Proposition 6.2).

Let $\pi:X\to Y$ be a flat projective family of n-dimensional varieties with central fibre $X_0$ be Calabi-Yau variety with canonical singularities then all general fibres $X_y$ are Calabi-Yau varieties with at worst canonical singularites. See Fibration when central fibre is a Calabi-Yau variety with canonical singularities

The inverse of this statement also holds true with some additional assumption.

Let $\pi:(X,D)\to Y$ be a flat projective family of n-dimensional varieties with $K_{X/Y}+D\cong \mathcal O_X(D)$ (i.e. fibres are log Calabi-Yau varieties) and $(X,(X_0,D_0))$ be divisorially log terminal then the log Calabi-Yau central fibre $(X_0,D_0)$ has at worst canonical singularities if and only if there exists an irreducible component $N$ of $X_0$ with a resolution of singularities $\tilde N\to N$ with $H^0(\tilde N,K_{\tilde N})\neq 0$. In fact, if $f:(\tilde X_0,\tilde D_0)\to (X_0,D_0)$ is a resolution of singularities, then we have

$$K_{\tilde X_0}+\tilde D_0\thicksim \sum_{E}a_EE$$ with $a_E\geq 0$, it follow that $K_{\tilde X_0}+\tilde D_0$ is effective. Now it is not very hard by the Idea of Wang to show that the converse is also holds true and we just need to know $K_0+D_0$ is irreducible and trivial.

By the results of Shigeharu Takayama, Wang and V.Tosatti if $0$ lies at finite Weil-Petersson distance then central fibre $X_0$ is Calabi-Yau variety with canonical singularities.

Note that for polarized Calabi-Yau degeneration $f:X\to \mathbb D$, the central fobre $X_0$ is Calabi-Yau variety with canonical singularities if and only if the Tian's Kahler potential of Weil-Petersson metric be upper bounded.

$$\omega_{WP}=-\sqrt[]{-1}\partial_y\bar\partial_y\log\int_{X_y}|\Omega_y|^2$$