Hashing large messages, how does it work?
While hashing, the messages are divided into chunks and padded so that a message is fit into the chunks, properly. For example in SHA-256 the message appended 1, then many zeroes and finally the message size in 64-bit added. The total size must be $x$ multiple of 512 where $x$ is minimum.

No, they are not separately hashed, they are chained. If not it how would you combine the output? If you don't combine what is a hash there? There is actually, hash trees famously know as Merkle Trees used to verify data blocks, or many files where you only keep the root hash.

SHA-256, for example, uses  Merkle–Damgård (MD) construction in which a compression function $F$ which is

$$F:\{0,1\}^{256}\times\{0,1\}^{512}\to\{0,1\}^{256}$$

The 512-bit input space is the message space. 256-bit input space is the previous hash, if it is the first block then an IV is used.

let $$M\mathbin\|pad=m_1\mathbin\|m_2\mathbin\|\ldots\mathbin\|m_\ell $$ $$H_i = F(m_i,H_{i-1})$$ with $H_0 = IV$ then

$$\operatorname{SHA-256}(M||pad) = F(m_\ell, F\ldots F(IV,m_1)\ldots))$$

This tells you about the MD construction, MD5, SHA-1, SHA-2 uses MD. There is also sponge construction that SHA-3 is based on, which left the OP as an exercise.