The red shift of distant stellar objects is supposed to be a corner stone of the big bang idea so is it true? Maybe not.
Of course we know that a redshift is caused by an object moving away from us at great speed. The same red shift is caused when an object escapes a gravity well, even leaving the Earth.
We are told that the rest of the universe is moving away from us (ignoring local gravity) due to the expansion of space, or rather the distance between us and everything else increasing since no one actually knows what space is (see my thread on the subject).
The thing to remember here is the speed of light. It is a constant. If the distance it travels through in one second expands by one nanometer due to expansion, light still travels at the same speed but takes a fraction of a second longer to travel that distance.
So it mounts up over distance and time. But not while you are watching (so making ridiculous the example of a train siren for the Doppler effect). It's (about) 14 miles per second over a million light years, so each second, one part in forty two million billion.
A photon does not get stretched by a ridiculously small amount of expansion as some claim. If such a totally insignificant force were capable of that, entering the Earth's gravity well would blue shift most photons to gamma rays.
It is said to happen over time but if you allow say a size of 10^-20 m for a photon, then each second it is expanding by just 1 part in 42x10^-36 of a meter. To put this in perspective, very energetic gamma rays go down to about 10^-15 m wavelength.
We are told that the light curve of distant super novae lengthens due to expansion but that idea only works up to around five billion light years. beyond that, no light curve from expansion, so the idea must be wrong. There is another cause for it.
And then there are linked quasars. Halton Arp's ideas so upset the BB community that because of their vitriol and sheer hatred he was forced to go to work in another country, but while some results may be explained away, others cannot and we have physically linked quasars with hugely different red shifts.