Everything works well with constraints which depend on time explicitly.
The definition of holonomic system of constraints is that, for $N$ material points represented in Cartesian coordinates in the rest space of a reference frame (the definition does not depend on the choice of the reference frame) a set of $c< 3N$ conditions must hold
$$f_j (t, \vec{x}_1, \ldots, \vec{x}_N)=0\quad j=1,\ldots,c\:, \tag{1} $$
where the functions f_j take values in $\mathbb{R}$ are $C^k$ with $k>1$ and, where (1) is valid these functions are functionally independent for every $t$. In other words, the Jacobian matrix of derivatives
$\frac{\partial f_j}{ \partial y_l}$ where $y_1,\ldots, y_{3N}$ are the components of the vectors $\vec{x}_1,\ldots, \vec{x}_N$, has rank $c$ when (1) holds.
In this case, locally, it is possible to describe every $\vec{x}_i$ as a function of $t$ and $n= 3N-c$ free components, denoted by $q^1,\ldots, q^n$ among the $y_r$. Different choices are also possible but, locally in space and time, the allowed configurations are defined by $n$ free coordinates.
If the constraints satisfy the physical requirement of ideal constraints regarding the behavior of the reactive forces, the Newton equations are equivalent to the usual Euler-Lagrange equations
written in terms of a curve in the space of the free coordinates.
The requirement of ideal constraint generalize the one of frictionless constraint including some further, physically relevant, possibilities as the rigidity constraint.
You see that there is no restriction on the time dependence of the functions $f_j$.
As an example, think of a curve $\Gamma= \Gamma(t, s) \in \mathbb{R}^3$, in the rest space of a reference frame, whose shape changes in time $t$ and where $s\in \mathbb{R}$ is a regular parameter along the curve. For instance
$$\Gamma(t,x) = (r+ ct^2) (\cos s {\bf e}_x + \sin s {\bf e}_y)\:.$$
where $r, c\in \mathbb{R}\setminus \{0\}$ are given constants.
This is a circle whose radius $R(t) = (r+ ct^2)$ depends on time.
This space can be obtained by imposing the constraints
$$f_1(t,x,y,z) := z \equiv 0$$
and
$$f_2(t,x,y,z) := (r+ ct^2)^2 - x^2 -y^2 \equiv 0\:.$$
It is not difficult to see the the associated Jacobian matrix has rank 2 where the two constrains are valid. Locally one can use the coordinate $x$ or the coordinate $y$ to describe the configuration space, but a better choice is to exploit an angular coordinate $s$ on the circle.
A material point of mass $m$ is constrained to stay on $\Gamma$ supposed to be frictionless.
There is no reference frame where this constraint does not depend on time.
We can use the parameter $s$ as Lagrangian coordinate. If there are no forces in addition to the reactive force due to the constraint, we can describe the equation of motion in terms of Euler-Lagrange equations of the Lagrangian
$$L(t, s, \dot{s}) = T(t,s,\dot{s})\:.$$
where
$$T = \frac{m}{2}{\bf v}(t,s, \dot{s})^2$$
and
$${\bf v}(t,s, \dot{s}) = 2ct (\cos s {\bf e}_x + \sin s {\bf e}_y)
+ (r+ct^2)\dot{s} (-\sin s {\bf e}_x + \cos s {\bf e}_y)\:.$$
So that
$$L(t, s, \dot{s}) = 4c^2t^4 +(r+ct^2)^2 \dot{s}^2 + (2 ct^2- (r+ct^2) \dot{s}) \sin (2s)\:.$$
The Euler-Lagrange equation for the curve $s=s(t)$ reads
$$\frac{d}{dt}\left(2(r+ct^2)^2 \frac{ds}{dt}
- (r+ct^2) \sin (2s) \right)- 2\left(2 ct^2- (r+ct^2) \frac{ds}{dt}\right) \cos (2s(t))=0\:.$$
