People here Pakistan, during the mourning processions during the months of Muharram and Safar, typically bring decorated horses as representations of the horse of Imam Hussain (as). They are called Shabeeh e Zuljanah. People do Ziyarah of them, and touch them, and express respect and love for them, and ask for the intercession of the Ahl ul Bayt (as) when they do so. Is this permitted? Is this a good practice?

There is no problem in doing. Different people need different ways of connecting to Imam Hussain (p) and feeling the tragedy of Karbala. As long as it does not introduce false things into religion or cause one to go astray, there is nothing bad in doing it.