About the Brownian bridge and its generalization (Markov)

Question

If $(B_t)$ is a standard Brownian motion, then

$$ B_t - \frac{T_1-t}{T_1-T_0} B_{T_0} - \frac{t-T_0}{T_1-T_0} B_{T_1}$$

is a Brownian bridge on $[T_0,T_1]$. It is know that this Brownian bridge is Markov.

Now consider another process

$$ U_t=A_t - \frac{T_1-t}{T_1-T_0} A_{T_0} - \frac{t-T_0}{T_1-T_0} A_{T_1}$$ where $(A_t)$ is an arbitrary continuous stochastic process.

1. If $(A_t)$ is Markov, does that imply that $(U_t)$ is Markov ?
2. If $(A_t)$ is Gaussian and Markov, does that imply that $(U_t)$ is Markov ?
3. If $(U_t)$ is Markov, does that mean $(A_t)$ is Markov ?
4. If $(U_t)$ is Gaussian and Markov, does that mean $(A_t)$ is Markov ?