Vtyjkyuk

Consider we have a mixture of multinomials or in a broader sense, a mixture of $f$s where $f$ is an distribution of exponential family type and the membership components are known with the sum of 1. Is the new distribution an exponential family too?

In general the answer is "no".

The mixture distribution
$$
alpha h_1(x)exp^theta_1^T phi_1(x) - A_1(theta_1) + (1-alpha) h_2(x)exp^theta_2^T phi_2(x) - A_2(theta_2)
$$
generally can't be factored into something like
$$
h_3(x) exp^theta_3^T phi_3(x) - A_3(theta_3)
$$
even if the two components come from the same exponential family (so $h_1=h_2, phi_1=phi_2, A_1=A_2$). A classic example is a mixture of Gaussians.

Although the product of exponential families is an (unnormalized)
exponential family, the mixture of exponential families is not an
exponential family.

--page 6 of Statistical exponential families: A digest with flash cards

In my viewpoint, this depends on the nature of the mixed distributions. Without loss of generality, consider a probability distribution with a single rate parameter $theta$. The latter probability distribution belongs to the exponential family if its PDF can be expressed in the form
$$
f(x | theta) = e^eta(theta)T(x)-A(theta)+B(x)
$$
Now, consider an exponential distribution with rate parameter $theta$ and a gamma distribution with known shape parameter $2$ and rate parameter $theta$ with corresponding PDFs
$$
f(x | theta) = theta e^-theta x equiv f(x | theta) = e^-theta x + log theta + 0
$$
and
$$
f(x | theta) = theta^2 x e^-theta x equiv f(x | theta) = e^-theta x + 2 log theta + x
$$
respectively. By mixing these distribution using the weights $theta(1 + theta)^-1$ and $(1 + theta)^-1$, respectively, we will get the PDF of the Lindley distribution that is given by
$$
f(x | theta) = theta^2 (1 + theta)^-1 (1 + x) e^-theta x equiv f(x | theta) = e^-theta x + log leftlbracetheta^2 (1 + theta)rightrbrace + log(1 + x)
$$

My personal verdict is that there might be "a few" special cases when a mixture of $f$s, where $f$ is a distribution of the exponential family, will yield a distribution that belongs to the exponential family.