Here are some inconclusive thoughts.

To construct (reasonable) limits of locally presentable categories, it suffices to consider [products, inserters, and equifiers](https://ncatlab.org/nlab/show/PIE-limit).

It's easy to check that locally $\kappa$-presentable categories are closed under small products. For colimits can be computed pointwise, and a generating set is given by objects whose components are $\kappa$-presentable, and initial except on a $\kappa$-small set of coordinates. If we were working with general $\kappa$-accessible categories, this would be more complicated -- I'm not sure, we might have to raise the degree of accessibility.

I suspect that when $\kappa$ is uncountable, locally $\kappa$-presentable categories are closed under inserters $Ins(F,G)$ between functors $F,G: \mathcal{C} \to \mathcal{D}$ which are cocontinuous and preserve the $\kappa$-presentable objects (equivalently, functors with a $\kappa$-accessible right adjoint). I don't have a complete proof. Colimits are computed as in $\mathcal{C}$, and it's easy to see that the objects whose underlying $\mathcal{C}$-object is $\kappa$-presentable are $\kappa$-presentable. It suffices to show that the colimit closure of these objects is all of $Ins(F,G)$ -- by Theorem 2.5.1 in Makkai-Paré, or by Lemma 3.7 in [these](http://home.sandiego.edu/~shulman/papers/generators.pdf) notes by Mike Shulman and the usual strong-generator characterization of locally $\kappa$-presentable categories. If $F(c) \overset{\gamma}{\to} G(c)$ is an object of $Ins(F,G)$, I would like to induct on the presentability rank $\lambda$ of $c$ to show that $F(c) \overset{\gamma}{\to} G(c)$ has the same presentability rank $\lambda$. The idea is that $c$ is a $\lambda$-small colimit of $\kappa$-presentable objects (see Adámek and Rosický Remark 1.30). If $\lambda = \mu^+$, then this means that $c$ is a $\mu$-sized colimit of $\kappa$-presentable objects. By adding $\mu$-small colimits to the indexing diagram, $c$ is in fact a $\mu$-sized, $\mu$-filtered colimit of $\mu$-presentable objects, and there must be a cofinal chain. So $c$ is the colimit of a $\mu$-sized chain of $\mu$-presentable objects $c = \varinjlim m_i$, which by induction are in the colimit closure we're interested in; moreover we can assume that the diagram functor preserves $\kappa$-filtered colimits. Now by $\mu$-presentability, for every $m_i$ the map $\gamma$ factors through $F m_i \to G m_{u(i)}$; we can assume $u(i) \geq i$. Look at the sequence $i, u(i), u(u(i)), \dots$. The $\kappa$th iterate must be a fixed point, so the fixed points form a cofinal set $J \subseteq I$. Then we can form $\varinjlim_{j \in J} (F m_j \to G m_j)$ as a colimit in $Ins(F,G)$ to complete the induction. The problem is that this induction only works when $\lambda$ is the the successor of a regular cardinal $\mu$ (I think it's fine if $\lambda$ is a regular limit cardinal, but it breaks down if $\lambda$ is the successor of a singular cardinal $\mu$ because the $\mu$-presentable objects don't actually have lower presentability rank than $c$ so the induction gets stuck -- the first issue is when $\lambda = \aleph_{\omega + 1}$), so the argument is not complete.

I haven't thought too much about equifiers, but I suspect that similar considerations are relevant as in the inserter case.